Higgs production via gluon fusion in association with two or three jets in supersymmetric models

Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN der Fakult¨at f¨ur Physik der Universit¨at Karlsruhe (TH)

genehmigte

DISSERTATION

von

Dipl.-Phys. Michael Kubocz aus Ruda Slaska

Tag der m¨undlichen Pr¨ufung: 06. November 2009 Referent: Prof. Dr. D. Zeppenfeld Korreferent: Prof. Dr. M. M¨uhlleitner

Abstract

Within the Minimal Supersymmetric Standard Model neutral Higgs bosons, the -odd Higgs A, a light and a heavy -even Higgs, h and H, can be CP CP efficiently produced via gluon fusion in high energy hadronic collisions. Real 4 5 emission corrections to Higgs production, at order αs and αs, lead to a Higgs plus two- and three-jet final state. This thesis presents the calculation of scattering amplitudes, as induced by triangle-, box-, pentagon- and hexagon- loop diagrams. Furthermore, the analytic expressions for the amplitudes were implemented into the Monte Carlo program VBFNLO, with which the numerical analysis was performed. Finally, resulting hadronic cross sections are discussed and phenomenologically relevant distributions for the Large Hadron Collider are shown.

Zusammenfassung

Innerhalb des Minimal Supersymmetrischen Standardmodells k¨onnen neu- trale Higgsbosonen, wie das -ungerade Higgsboson A, das leichte und CP schwere -gerade Higgsboson h und H, effizient durch Gluonfusion in hoch- CP energetischen hadronischen Kollisionen erzeugt werden. Reelle Emissions- 4 5 korrekturen zur Higgsproduktion der Ordnung αs und αs, f¨uhren zu einem Endzustand mit zwei oder drei Jets. In dieser Dissertation wird die Berech- nung von Streuamplituden, induziert durch Dreieck-, Box-, Pentagon- und Hexagonschleifendiagramme pr¨asentiert. Desweiteren wurden die analytis- chen Ergebnisse in das Monte-Carlo Programm VBFNLO implementiert und mit dessen Hilfe die numerische Auswertung durchgef¨uhrt. Anschliessend werden resultierende hadronische Wirkungsquerschnitte diskutiert und ph¨a- nomenologisch wichtige Verteilungen f¨ur den LHC gezeigt.

Contents

1 Introduction 3

2 Basics 7 2.1 Standard Model and Higgs Mechanism ...... 7 2.2 Higgs mechanism in the MSSM ...... 10 2.3 Masses and mixing patterns of squarks ...... 13 2.4 QuantumChromo-Dynamics...... 14

3 Higgs production processes in association with two and three jets 17 3.1 Introductionandmotivation ...... 17 3.2 Outlineofthecalculation ...... 19 3.3 Higgs production in association with two jets ...... 22 3.3.1 qQ qQΦ and qq qqΦ ...... 23 → → 3.3.2 qg qgΦ ...... 25 → 3.3.3 gg ggΦ ...... 27 → 3.4 Higgs production in association with three jets ...... 34 3.4.1 qQ qQgΦ and qq qqgΦ...... 37 → → 3.4.2 qg qggΦ...... 40 → 3.4.3 gg gggΦ ...... 46 → 4 Numerical implementation and checks 49

5 Applications to LHC physics 53 5.1 Introduction...... 53 5.2 Production of the -odd Higgs boson A in association with CP twojets ...... 54 5.3 Production of a violating Higgs boson Φ in association CP withtwojets ...... 59 4 CONTENTS

5.4 Production of the -even Higgs bosons h and H in associa- CP tion with two jets with additional squark contributions . . . . 61 5.5 Production of the Higgs bosons Φ in association with three jets 67

6 Conclusions 71

A Higgs vertices to fermions and sfermions 73 A.1 Higgscouplingstofermions ...... 73 A.2 Higgscouplingstosfermions ...... 75

B Loop tensors 79 B.1 Three-point functions (Triangles) ...... 79 B.1.1 Fermion-triangle with Higgs vertex ...... 79 B.1.2 Fermion-triangle with -oddHiggsvertex...... 80 CP B.1.3 Fermion-triangle with -even Higgs vertex ...... 81 CP B.1.4 Sfermion-triangle with -even Higgs vertex ...... 82 CP B.1.5 Sfermion-triangle with -even Higgs vertex andq ˜qgg˜ - CP vertex ...... 84 B.1.6 Sfermion-triangle with -even Higgs vertex and two CP q˜qgg˜ -vertices...... 85 B.2 Four-pointfunctions(Boxes) ...... 86 B.2.1 Fermion-box with Higgs vertex ...... 86 B.2.2 Fermion-box with -oddHiggsvertex ...... 88 CP B.2.3 Fermion-box with -evenHiggsvertex ...... 89 CP B.2.4 Sfermion-box with -evenHiggsvertex ...... 90 CP B.2.5 Sfermion-box with -even Higgs vertex andq ˜qgg˜ -vertex 91 CP B.3 Five-point functions (Pentagons) ...... 94 B.3.1 Fermion-pentagon with Higgs vertex ...... 94 B.3.2 Sfermion-pentagon with -even Higgs vertex . . . . . 95 CP B.4 Six-point functions (Hexagons) ...... 96

C SU(N) identities 99 C.1 SU(N)tensors...... 99 C.2 Tracesofcolorgenerators ...... 99 C.3 Convolutions of da1a2a3 and f a1a2a3 with ta ...... 100 C.4 Jacobiidentities...... 100 D QCD- and SQCD Vertices 101 D.1 QCDvertices ...... 101 D.2 SQCDvertices(onlysquarks) ...... 102

E Effective Lagrangian 103

F gg → gggΦ 107 F.1 Remaining color factors for diagrams with pentagons . . . . . 107

Bibliography 111

Acknowledgements 117

1

Chapter 1 Introduction

In today’s modern physics, the process of gaining knowledge proceeds in a close cooperation of theory and experiment. By empirical research, claiming objectivity and reproducibility of the observation, theoretical assumptions are verified. Afterwards, the empirical models of experimental physics are attributed in a mathematical way to the known theoretical basis. Or, if that is not possible, a new theoretical model has to be invented, which ful- fills the experimentally measured data. However, a theoretical model should not only reproduce experimental data, but also make predictions beyond the experimentally investigated region of the nature or the considered sys- tem respectively, which have not yet been discovered. With the discovery of quantum mechanics (QM) and its improved development to quantum field theory (QFT), the proof of renormalizability of non-Abelian gauge theories by G. ’t Hooft and Veltman, the basic building blocks were set to the best experimentally verified and theoretically well established model, applicable over a wide range of conditions, the Standard Model (SM). The Standard Model is based on the gauge group SU(3) SU(2) U(1) C ⊗ L ⊗ Y and provides a unified framework to describe phenomena of the electro-weak as well as of the strong interaction. In spite of the successful description of elementary particle physics, the SM still has some crucial problems, which spoil the elegancy of this theoretical framework. It starts already with the absence of gravity. But, ok, for this problem one can turn a blind eye, because gravitational effects can be safely neglected at the level of elemen- tary particle interactions, due to very tiny effects. The introduction of a scalar field, whose vacuum expectation value breaks the gauge symmetry SU(2) U(1) U(1) spontaneously to generate masses for leptons, L ⊗ Y → Q quarks and electro-weak gauge boson, seems to be a smart step for the first

3 glance. These masses are generated in a natural way without spoiling gauge invariance and other symmetries of the theory, and only one new particle, the Higgs boson, has to be found in experiments at particle colliders. However, to this day neither the LEP nor the Tevatron collider has found any evidence for the Higgs boson. But, they could provide a lower limit of 114.4 GeV for the SM Higgs boson mass. Unfortunately, the introduction of this scalar particle introduces a further problem, the so called ”hierarchy problem”. To guarantee unitarity of the scattering amplitude of electro-weak gauge bosons, the Higgs mass cannot be larger than 1 TeV. Though, quantum corrections spoil this limit, since they are proportional to a scale, which can be arbitrary large. A further problem of the SM is the extrapolation of gauge couplings to a high energy scale using methods of the renormalization group, that, however, does not lead to a unification. Furthermore, the absence of de- scribing and providing candidates for dark matter does not make a lasting impression on the SM... Roughly speaking, there is still enough space for extensions. Many interesting extensions are already available on the mar- ket of elementary particle physics: extended Techni-color models, models with extra dimensions, Little Higgs models,... and of course supersymmetric models, like the minimal supersymmetric extension of the SM, the MSSM. It features the extended Higgs sector of a two-Higgs-doublet model: a light and a heavy neutral -even Higgs, h and H,a -odd Higgs A and two charged CP CP Higgs bosons H±, and, of course, additional particles, which are referred as super-partners to the particle content of the SM. It attracted attention due to possible candidates for dark matter, and the extended particle content provides solutions to the above mentioned hierarchy problem. In addition, the much more constrained Higgs sector of the MSSM sets a limit for the mass of the light scalar h to m . 135 GeV, which points towards m 120 h h ≈ GeV at the 95 % CL. The search for the Higgs bosons of the SM and its extensions, as well as the investigation of their couplings, is the main mo- tivation for the construction of the Large Hadron Collider (LHC) at CERN with 14 TeV center-of-mass energy . The experimental evidence of Higgs bosons would confirm the idea of electro-weak unification and also show in- dication for theory beyond the SM. The most copious sources of the Higgs boson production at the LHC are electro-weak boson fusion and gluon fusion. Higgs production via fusion of

4 electro-weak gauge bosons in association with two tagging jets provide an adequate possibility to measure the Higgs boson couplings. Small radia- tive corrections and, hence, small systematic errors characterize this process. Gluon fusion gives also rise to a Higgs plus two jet final state and constitutes a large background, which cannot be neglected. Furthermore, in gluon fusion the azimuthal angle φjj between the more forward and more backward of the two tagging jets is sensitive to the -character of the Higgs coupling to CP fermions. The subjects of this thesis are the calculation and the discussion of the phe- nomenological implications on the production of neutral Higgs bosons with two and three additional jets in final state via gluon fusion in high energy hadronic collisions at the LHC. These are processes like pp Higgs + jjX → and pp Higgs + jjjX. As theoretical basis, the minimal supersymmetric → extension of the Standard Model (MSSM) is used. In the following, charged Higgs bosons are not considered and all neutral Higgs bosons are denoted by the shorthand Φ. The contributing sub-processes to the Φjj and Φjjj pro- duction include quark-quark, quark-gluon and gluon scattering. At leading order, the coupling of the neutral Higgs bosons to gluons is mediated by a massive loop, whose particle content is restricted in this thesis to the third generation of quarks and its supersymmetric scalar partners, the squarks. Thus the requirement of a massive loop is justified, since the Higgs bo- son couples only to massive particles and the gluon, carrier of the strong force, has no mass. The loop-topologies, appearing in the calculation of the amplitudes, range from triangles, boxes, up to pentagons and hexagons, which make the calculation quite involved. Additionally, Quantum Chromo- Dynamics (QCD) provides a complicated color structure for the scattering amplitudes. First results of double real emission corrections to the produc- 4 tion of the HSM, which lead to a Higgs plus two-jet final state at order αS, were already presented in Refs. [1, 2]. A further calculation with the same final state was performed for the -odd Higgs boson A and is available in CP Ref. [3]. The results of the calculation, presented in this thesis, were implemented into the gluon fusion part GGFLO of the parton-level Monte Carlo program VBFNLO [4]. The thesis is organized as follows:

Chapter 2 gives a brief outline of the Standard Model and its particle content,

5 as well as the Higgs mechanism in the SM and in the supersymmetric exten- sion, the MSSM. Furthermore, the masses and mixing patterns of squarks are preparatively introduced. And finally, the basics of perturbative QCD and the calculation of the cross section in hadronic collisions are reviewed.

Chapter 3 provides a detailed description of the calculation of the scattering amplitudes for the two processes pp Higgs + jjX and pp Higgs + jjjX. → → Further details on various loop contributions are relegated to the Appendices.

Chapter 4 describes analytical and numerical consistency checks of the cal- culation and implementation.

The main phenomenological results are presented in Chapter 5, for pp scat- tering at the LHC with a center of mass energy of √s = 14 TeV. Integrated cross sections and differential distributions for the production of the -odd CP Higgs boson A are shown, as well as arbitrary -violating couplings to the CP third generation of quarks for general Φjj events. Furthermore, differential distributions and integrated cross sections are presented for the production of the -even Higgs bosons h and H including fermionic and scalar loop- CP contributions of the third generation of quarks and squarks. The last part provides preliminary results for Φjjj production, showing de-correlation ef- fects of the azimuthal angle distribution, in comparison to Φjj events.

Final conclusions are drawn in Chapter 6.

Vertices of the neutral Higgs bosons to fermions and sfermions are given in Appendix A. The Appendix B provides a detailed description of the cal- culated and implemented loop topologies. In Appendix C useful identities of the SU(N)-algebra are collected. Vertices of the QCD and SUSY-QCD (SQCD) are illustrated in Appendix D. The Appendix E gives a short intro- duction to the effective Lagrangian and its correspondence to the full theory. Finally, Appendix F contains some stored color structures of the sub-process gg gggΦ. →

6 Chapter 2 Basics

2.1 Standard Model and Higgs Mechanism

The Standard Model is based on some basic principles: special relativity, locality, quantum mechanics, renormalizability, local and global symmetries. The model provides a unified framework to describe three forces of nature: the electro-weak theory, proposed by Glashow, Salam and Weinberg. It de- scribes electromagnetic and weak interactions between lepton and quarks. The underlying group structure is composed of a direct product of two gauge symmetry groups SU(2) U(1) , corresponding to weak left-handed isospin L ⊗ Y and hypercharge. The theory of strong interactions between colored quarks enters via Quantum Chromo-Dynamics (QCD), which is based on the non-

Abelian gauge symmetry group SU(3)C . Thus, the full group structure reads as

SU(3) SU(2) U(1) . (2.1) C ⊗ L ⊗ Y The content of matter fields is given by three generations of chiral leptons and quarks. They are split to left-handed isospin doublets and right-handed isospin-singlets: Leptons: • ν ν ν L = e , µ , τ ; L = e , µ , τ , L e µ τ R R R R !L !L !L Quarks: • u c t Q = , , ; Q = u , d , c , s , t , b . L d s b R R R R R R R !L !L !L

7 The hypercharge of fermions Yf is defined by the Gell-Mann-Nijishima rela- tion

Y =2 Q I3 , (2.2) f f − f   3 with Qf the electric charge in units of +e, and If , the third component of the weak isospin. The values of Yf for all generations of the fermion matter fields are given by

1 4 2 Y = 1,Y = 2,Y = ,Y = ,Y = . (2.3) L − eR − Q 3 uR 3 dR −3

Leptons do not carry color charge and, hence, are color-singlets under the global SU(3)-symmetry of the QCD, in contrast to quarks, which are assigned as SU(3)-triplets. Forces between fermionic matter fields are mediated by spin-one bosons. These vector fields are derived assuming, that the Lagrangian of a fermion field is invariant under local gauge transformations for a group G. Here, of course, the group G corresponds to the three sub-groups, mentioned in Eq. (2.1). Carrier of the electromagnetic force is the massless photon γ, whereas the massive W and Z bosons mediate the weak force between the SU(2)-multiplets, described above. The neutral gauge boson Z interacts with all fermion fields, the charged gauge bosons W ±, however, induce transitions between the left-handed doublets only. The right-handed singlets remain untouched. In addition, with this different coupling behavior to left- and right-handed fermions, the experimentally observed violation of parity of the weak force is considered automatically. The strong force between quarks is mediated by eight massless gluons, which carry simultaneously color and anti-color. The experimentally observed masses of the weak gauge bosons, leptons and quarks turn out to be problematic from the theoretical point of view. Explicit 2 µ mass terms for gauge bosons, m A Aµ, destroy gauge invariance, which is a basic requirement for the renormalizability of the theory. Mass terms for lep- tons and quarks, mF F = m F LFR + F RFL , are also not possible, because the product of a doublet with a singlet is not invariant under SU(2) -isospin
L rotations. This problem can be circumvented, if the model is extended by a

8 further scalar SU(2)-doublet field Φ

φ+(x) Φ(x)= 0 , (2.4) φ (x)! with hypercharge YΦ = 1 and potential

λ 2 V (Φ) = µ2 Φ†Φ + Φ†Φ , with µ2 > 0,λ> 0 . (2.5) − 4

The minimization of V breaks the SU(2) U(1) U(1) spontaneously L ⊗ Y → Q and the scalar field Φ acquires a VEV v

1 0 2µ −1/2 Φ = with v = = √2GF 246 GeV . (2.6) h i √2 v! √λ ≈  

GF denotes the Fermi-constant with the experimentally fixed value at GF = 1.16637 10−5. The expansion of Φ around the VEV can be written as ·

1 θ1(x)+ θ2(x) Φ(x)= 0 . (2.7) √2 v + HSM (x) + iθ3(x)! According to the Goldstone Theorem, the number of Goldstone bosons,

θ1,2,3(x), corresponds to the number of broken generators. Furthermore, they can be rotated away using the local SU(2)Y symmetry (unitary gauge)

τ a 1 0 Φ(x) exp i θa Φ(x)= . (2.8) → − v √ v + H0 (x)   2 SM ! Masses of the gauge bosons are given by the the kinetic term Y D Φ † DµΦ with D Φ= ∂ ig W aτ a ig B Φ , (2.9) µ µ µ − 2 µ − 1 2 µ   a
a
a in which τ = σ /2 with a = 1, 2, 3 are the known Pauli-matrices. Wµ correspond to the gauge bosons and g2 to the coupling constant of the SU(2)L group. Bµ and g1 describe the U(1)Y gauge boson and its coupling constant. After the diagonalization procedure, one gets the following relations for the masses of the gauge bosons v v M = g2 + g2 , M = g , M =0 . (2.10) Z 1 2 2 W 2 2 Aµ q 9 Since the SU(3) symmetry is not broken, there is no mass term for the gluon. Now, the Higgs-doublet Φ allows for Yukawa-like terms

g (F Φ)F + h.c. , (2.11) f L · R h i that generate, after the spontaneous symmetry breakdown, the wanted mass- terms for fermions v gf f fR + f fL , (2.12) √2 L R
where f denotes the entry of the corresponding SU(2) doublet and singlet. Finally, the insertion of the Higgs-doublet (2.8) into the potential (2.5) gives the mass of the Higgs boson v MH0 = √2µ = √λ . (2.13) SM √2

2.2 Higgs mechanism in the MSSM

The Minimal Supersymmetric Standard Model (MSSM) is the simplest su- persymmetric extension of the SM. In this connection, supersymmetry ex- tends the Poincar´e-group by new generators, that correlate bosonic and fermionic degrees of freedom. For a more introductive description see Ref. [5]. Thus, the particle spectrum of the SM gets expanded. Quarks and leptons obtain scalar partners, called squarks and sleptons. The supersymmetric partners of the gauge bosons are spin-1/2 particles, photinos, winos, zinos and gluinos. Superpartner of the Higgs boson is the higgsino. The particles of the SM and their super-partners are then combined to superfield multiplets. In contrast to the SM, the MSSM contains two Higgs-doublets

0 + φ1 φ2 Φ1 = − ,YΦ1 = 1 , Φ2 = 0 ,YΦ2 =1 . (2.14) φ1 ! − φ2 !

The requirement of two doublets with opposite hypercharge is necessary to compensate the hypercharges of both higgsinos. This guarantees, that the sum of all hypercharges vanish, similar to the SM, and leads to an anomaly

10 free model. Yukawa couplings to both Higgs-doublets are given by the su- perpotential [6]

W = Y uuˆ Φˆ Qˆ + Y ddˆ Φˆ Qˆ + Y l lˆ Φˆ Lˆ − ij Ri 2 · j ij Ri 1 · j ij Ri 1 · j i,jX=Gen. + µ Φˆ Φˆ , (2.15) 2 · 1 where the hat marks the corresponding superfield multiplets and the Y de- notes Yukawa couplings among lepton and quark generations. The mass parameter µ is necessary to avoid an additional Peccei-Quinn symmetry [7]. Moreover, supersymmetry claims for the superpotential W to be a holomor- phic function with respect to the superfields and, thus, forces the Higgs- doublet Φ1 to couple to down-type fermions and Φ2 to up-type fermions only. By the way, this different coupling behavior of both Higgs-doublets corresponds exactly to that of the general 2HDM Model II [8]. The scalar potential reads as

V = m2 φ0 2 + φ− 2 + m2 φ0 2 + φ+ 2 m2 φ−φ+ φ0φ0 H 1 | 1| | 1 | 2 | 2| | 2 | − 3 1 2 − 1 2 2 2 g + g

2 + h.c. + 1 2 φ0 2 + φ− 2 φ0 2 φ+ 2 8 | 1| | 1 | −| 2| −| 2 | g2

+ 2 φ−∗φ0 + φ0∗φ+ 2 , (2.16) 2 | 1 1 2 2 | with the abbreviations

m2 = µ 2 + m2 , m2 = µ 2 + m2 , m2 = Bµ , (2.17) 1 | | Φ1 2 | | Φ2 3 containing soft SUSY-breaking scalar Higgs-mass terms mΦi [6]. The con- stants g and g are the usual SU(2) U(1) gauge couplings. The symmetry- 2 1 L⊗ Y breaking proceeds similar to the SM. Here, the neutral components of both Higgs fields acquire VEVs

1 v1 1 0 Φ1 = , Φ2 = , (2.18) h i √2 0 ! h i √2 v2! with

v2 v sin β 2 2 2 tan β = = and v = v1 + v2 246 GeV . (2.19) v1 v cos β ≈

11 Phase freedom of both VEVs restricts the angle β to the range 0 β π/2. ≤ ≤ The expansion of Φ1,2 around the VEVs yields

0 0 + 1 v1 + H1 + iP1 1 H2 Φ1 = − , Φ2 = 0 0 . (2.20) √2 H1 ! √2 v2 + H2 + iP2 ! The mass-matrix of the components of both Higgs-fields results from the second derivatives of VH (2.16), according to the corresponding fields at the points given in Eq. (2.18). The conservation of electric charge avoids mix- ing between charged and neutral Higgs fields. Consequently, there are two mass matrices, one for the charged sector and one for the neutral sector [9]. Moreover, assumed -invariance of the Higgs sector does not mix real and CP imaginary components of the neutral Higgs bosons. The eight degrees of freedom correspond then to three Goldstone bosons G0, G± and five Higgs bosons, neutral h, H, A and also two charged H±. The light h and heavy H are -even, whereas G0 and A are -odd. The diagonalization of the CP CP mass-matrix is done with the following transformations

G0 cos β sin β P 0 = 1 , (2.21) A sin β cos β P 0 ! − ! 2 ! G± cos β sin β H± = 1 , (2.22) H± sin β cos β H± ! − ! 2 ! H cos α sin α H0 = 1 , (2.23) h sin α cos α H0 ! − ! 2 ! with β given in Eq. (2.19). The angle α is defined in the following way

m2 + m2 m2 + M 2 tan2α = h H tan2β = A Z tan2β , (2.24) m2 M 2 m2 M 2 A − Z A − Z in which α is restricted to the interval π/2 α 0 by 0 β π/2. The − ≤ ≤ ≤ ≤ tree-level physical squared masses of the two -even Higgs bosons are CP 1/2 2 1 2 2 2 2 2 2 2 2 mH,h = mA + MZ mA + MZ 4MZ mA cos 2β . (2.25) 2( ± − ) h
i Hence, the Higgs-mass spectrum is completely controlled at tree-level by mA and tan β. Further important tree-level relations for the remaining Higgs-

12 masses are [10]

2 2 2 2 2 mH± = mA + MW > max MW , mA , (2.26) 2 2 2 2 mh + mH = mA + MZ ,
(2.27) m < min m , M cos2β < min m , M , (2.28) h A Z | | A Z mH > max mA, MZ
.
(2.29)

Furthermore, radiative corrections
shift the tree-level masses to higher values. These corrections can be very large, because they contain couplings to top- quarks as well as to their scalar super-partners. Especially the mass of the light Higgs boson h is increased by several tens of GeV, shifting it from the

MZ value to mh . 135 GeV. The corresponding correction, which contain top-quark and -squark contribution is given by 3 ∆m2 = m4 log m2/m2 . (2.30) h 2π2v2 t t˜ t
For a more detailed description of the Higgs-sector in the MSSM see Refs. [6, 9, 10]. Couplings of Higgs bosons to fermions are, in comparison to the SM, modified by trigonometric functions containing the angles α and β. They can be looked up in App. A. Moreover, W W A, ZZA and WZH± vertices are not present at tree-level, because they are forbidden by -invariance. CP

2.3 Masses and mixing patterns of squarks

There are three sources of squark-mass terms in the scalar potential part of the Lagrangian density . They are given by

q˜ q˜ q˜ q˜ V = VSOFT + VF + VD . (2.31)

q˜ VSOFT contains explicit mass-terms as well as trilinear A-terms. F-term con- tributions arise from the superpotential W (2.15). Contributions of the D-terms are specified in Eq (2.33). The squared mass-matrix for up-type squarks can be then written as

2 2 ˜ u m˜ + mt + D tL mt A + µ cot β tL − , (2.32) u 2 2 ˜ mt A + µ cot β m˜ + mt + D tR −
tR
!

13 with 1 2 D t˜ = M 2 cos2β sin2 θ , L Z 2 − 3 W
 2  D t˜ = M 2 cos2β + sin2 θ . (2.33) R Z 3 W   The eigenvalues
of this matrix reads as follows

2 1 2 2 1 2 2 m˜ = m˜ + m˜ + M cos2β + m t1,2 2 tL tR 4 Z t
2 1 2 2 2 1 2 2 mt˜ mt˜ + MZ cos2β sin θW ∓ ( 2 L − R 4 − 3    1 2 2 t 2 mt µ cot β + A . (2.34) − )
If one assumes no flavor mixing, the top-squark mixing matrix has the fol- lowing form

t˜ cos θ sin θ t˜ 1 = t t L . (2.35) t˜ sin θ cos θ t˜ 2! − t t! R!

The mixing angle θt can be deduced from

2 2 2 1 2 2 2 m˜ + mt + MZ cos2β sin θW m˜ tan θ = tL 2 − 3 − t1 . (2.36) t m (At + µ cot β) − t

Taking into account that m˜bL = mt˜L via SU(2) symmetry, the derivation of the mass-matrix for down-type squarks follows the same procedure, but with corresponding definitions for charges, angles and masses (see Ref. [9]). The off-diagonal terms in the matrix of Eq. (2.32) can be particularly large for the third generation of squarks. In the case of the top-squark, these terms are proportional to the top-quark mass and yield a large mass splitting between t˜1 and t˜2, which can make t˜1 very light. Higgs couplings to up- and down-type squarks can be found in App. A.

2.4 Quantum Chromo-Dynamics

QCD is a gauge theory, based on the local non-Abelian SU(3)C group and describes interactions of the strong force between quarks, the constituents

14 of hadrons. The involved charge of the interaction is called color and cor- responds to three degrees of freedom of the fundamental representation of the global SU(3) symmetry. The strong force is mediated by eight spin-1 bosons, the gluons, which carry color and anti-color. They are related to the conjugated representation of the SU(3) symmetry group. Due to the fact, that gluons are charged, in contrast to the photon of QED, they can interact also among each other via three- and four-gluon vertices. This property has 2 an effect on the coupling strength αS, which is defined at a certain scale Q in the following way

2 2 2 gS Q αS Q = 4π

α (µ2) = S . (2.37) 1+ α (µ2)/12π(11N 2N ) log(Q2/µ2) S C − f

Nc denotes the number of colors, Nf the number of fermions and µ is a ref- erence scale. For Q2 the coupling strength α vanishes. This behavior →∞ S is known as asymptotic freedom and was discovered by Wilczek, Gross and Politzer [11, 12]. Now, this feature gives the possibility to perform pertur- bative QCD calculations at high energies. Whereas for low energies, calcu- lations get more involved and cannot be done perturbatively. At a scale of 200 MeV, called Λ , α even diverges. ≈ QCD S Moreover, in experiments, no colored particles were found, but colorless bound-states. This phenomenon is called confinement and is also an effect of the gluon self-coupling. For more details about QCD see e.g. Refs.[13, 14].

The analytic expression for a hadronic cross section with two partons (quarks and gluons) a1 and a2 in initial state reads as follows

σ = dx1dx2 fa1 (x1,µF ) fa2 (x2,µF ) Z sub-procX 1 2 1/2 dΦΘcuts sub-proc , (2.38) × 4 (p p ) m2 m2 |M | a1 · a2 − a1 a2 Z  1/2 where the flux-factor (p p ) m2 m2 = 2E 2E v v states a1 · a2 − a1 a2 a1 a2 | a1 − a2 | the relative velocity of the beams. The shortcut 2 denotes the    |Msub-proc|  squared expression of the Feynman amplitude for different sub-processes,

15 summed over final polarizations and colors and averaged over initial polar- izations and colors. The explicite expression is given by

2 1 1 2 sub-proc = sub-proc . (2.39) |M | 4 #(color)(a1, a2) |M | colorX Xpol.

The function Θcuts describes selection cuts, which are applied on the kine- matics of particles in final state. These cuts restrict the integration over the phase-space to regions, that are accessible by experiments. Furthermore, they anticipate detector capabilities and jet finding algorithms. The general expression of the Lorentz-invariant phase-space for a n-particle final-state is defined as

n d3p dΦ (P ; p ,...,p )= i (2π)4δ4 P p . (2.40) n 1 n (2π)32E − i i=1 i ! i ! Y X The final result is then obtained by the twofold integration over the parton density functions (PDFs) fai . A PDF describes the probability density to find a parton ai inside a nucleon with a certain longitudinal momentum fraction between xi and xi + dxi. The parameter µ is called factorization scale and corresponds to the resolution, at which short- and long-distance physics is separated. If all orders of a perturbation series are taken into account, the final result for a physical observable would not depend on µ. But, in a finite order perturbation theory physical observables involve the parameter µ, which is then set to the characteristic scale of the underlying process.

16 Chapter 3 Higgs production processes in association with two and three jets

3.1 Introduction and motivation

In pp-collisions at the LHC several mechanisms, depicted in Fig. 3.1, con- tribute to the production of the Higgs boson in the SM. The dominant pro- duction mechanism, over the entire Higgs mass range and accessible at the LHC, is gluon fusion (a), which is mediated mainly by a top-quark loop. The corresponding cross section, including all top-mass effects, is also known at NLO [15, 16]. Further NNLO calculations were done only within the effective Lagrangian approximation (see App. E), where the top-mass dependence is integrated out by using the limit m [17, 18]. The second largest cross t →∞ section is produced via weak boson fusion (b), mediated by t-channel W or Z exchange. It contains an additional pair of hard tagging jets in the final state, arising from scattered quarks. In addition to this, the same signature can also appear in gluon fusion via (α2 ) real emission corrections [2]. They O S are discussed and described in detail in the following sub-chapters. QCD corrections to weak boson fusion [19, 20] are known to NLO and show the expected improvement of scale uncertainties for cross sections and different distributions. Full NLO QCD corrections to the Hjj production process via gluon fusion are only available in the effective Lagrangian approximation [21]. Since the lowest order process is loop-induced, a full NLO calculation would entail a two-loop evaluation, which is presently not feasible. In this thesis, only real emission corrections with full-mass effects are discussed in the sub-

17 X q

p p g t H V H

p p q X

(a) (b)

t p p g q V H H q¯ p p t¯

(c) (d)

Figure 3.1: Higgs production mechanisms at LHC: (a) Gluon fusion, (b) weak boson fusion, (c) radiation of a Higgs boson off top pair, (d) Higgs production in association with a W or Z boson.

sequent chapters. Virtual contributions, coming from two-loop topologies, are not considered here, but pentagon- and hexagon-loops, which complicate the evaluation, especially for Hjjj. Further production mechanisms are the radiation of a Higgs boson off top pairs (c) [22, 23] and electro-weak boson associated production (d) [24, 25], where the involved boson is identified by its leptonic decay. In the MSSM gluon fusion is again the most important production channel for the Higgs bosons h, H and A. Also Higgs-Strahlung off top and, especially, bottom quarks, which are enhanced by tan β, play an important role. Higgs radiation off W or Z bosons and Higgs production via weak boson fusion is only interesting in association with the light Higgs

18 boson h. The next chapter describes the calculation of Higgs production processes with two and three additional jets. In the following, additional contributions with squarks are taken into account only for the process with a two jet final state. Loop-topologies, containing gluino-vertices, are neglected, due to the fact, that these contributions are suppressed by large gluino-masses on the one hand. And on the other hand, they appear in loop-diagrams with through- going light quark lines and, hence, provide an additional suppression of the Higgs coupling to squarks. Furthermore, no ghost-fields c and c appear in the calculation, because the colored QCD-ghosts are massless and therefore imply a vanishing coupling to the Higgs bosons.

3.2 Outline of the calculation

Due to a large number of contributing Feynman graphs, it is most convenient to give analytic results for the scattering amplitudes for fixed polarizations of external quarks and gluons, instead of using trace techniques to express polarization averaged squares of amplitudes in terms of relativistic invariants. The spinor algebra of those helicity amplitudes can be handled easily with the help of the formalism and methods developed in Ref. [26, 27]. Further calculation methods are based on the framework introduced in Ref. [1, 2]. The analytic expressions of amplitudes are then evaluated numerically with the gluon fusion part GGFLO of the program VBFNLO [4]. Throughout the calculation all masses of external fermions are set to zero. If not specified explicitly, all gluon momenta are treated as outgoing. Working in the chiral representation, the external (anti-) fermions can be described by a two-component Weyl-spinor of chirality τ [27]

0 ψ (pi, σi)τ = Si 2pi δσiτ χσi (pi) . (3.1) q Here, pi denotes the physical momenta expressing the phase space and the wave functions of fermions of fixed helicity σi, as well as qi for on-shell gauge bosons, whereas pi and qi describe momenta appearing in the momentum flow in Feynman diagrams. Both sets of momenta and helicities are related by the sign factors Si

pi = Si pi , qi = Si qi and σi = Si σi , (3.2)

19 with S = +1 for fermions and S = 1 for anti-fermions. Momenta q of i i − i out- and ingoing gluons are controlled by the factors Si = +1 and Si = 1 respectively. This allows easy switching between different production − channels. The factor δσiτ ensures conservation of helicity in the chiral limit and χσi (pi) denotes the helicity eigenstates. The polarization vectors for on-shell gluons are available in VBFNLO [4] in two different representations [28], which, of course, satisfy the transversality property q ǫ(q) = 0 with ǫµ(q)= 0,~ǫ (q) : (3.3) · ∗ 1) real polarization vectors, i.e. ǫi (qi)=
ǫi(qi), with gauge boson momen- tum qµ as basis vector: ǫ (q , 1) = ~q q −1 0, q q , q q , q2 , (3.4) i i | i| iT ix iz iy iz − iT ǫ (q , 2) = q −1
0, q , q , 0
(3.5) i i iT − iy ix with

qµ = E , q , q , q , E = ~q 2 + m2 1/2 , (3.6) i i ix iy iz i | i| 2 2 1/2

qiT = qix + qiy . (3.7) These real polarization
vectors describe transverse polarized gluons. 2) Complex polarization vectors in the helicity basis λ = : ± 1 ǫi(q ,λ = )= ( ǫi(q , 1) iǫi(q , 2)) . (3.8) i ± √2 ∓ i − i Furthermore, through-going fermion lines with attached gluon vertex and propagator are combined to effective quark currents, 1 1 J µ = δ χ† (¯p )(σµ) χ (¯p ) = δ f (σµ) i , (3.9) fi σf σi σf f τ σi i q2 σf σi h | τ | i(p p )2 fi i − f µ where σ τ=± are the Dirac matrices in the two-component Weyl-basis, i , f , abbreviations for in- and outgoing external fermions and τ = σ = σ | i h |
i f denotes helicity conservation. Analogously to that, one can define a gluon- current, which is composed of a three-gluon-vertex contracted with two po- larization vectors of outgoing on-shell gluons with indices i, j

G,µ 1 Jij (qij)= 2 (qi + qj) µ 2ǫ(q )µ q ǫ(q )+ ǫ(q ) ǫ(q ) q q 2ǫ(q )µ q ǫ(q ) . (3.10) × i i · j i · j j − i − j j · i h
i 20 The propagator term belongs to the uncontracted virtual gluon with momen- tum qij = qi + qj, which is here defined as ingoing. In addition to that, the transversality property of Eq. (3.3) was used. For the emission of an on-shell gluon close to an external quark, one can use a Bra and Ket notation similar to the quark current of Eq. (3.9), but with an additional momentum variable ql of the corresponding polarization vector [26] 1 q , i =(p q ) (ǫ ) χ (¯p ) , (3.11) | l i i − l −σi l σi σi i (p q )2 i − l 1 f, q = χ† (¯p )(ǫ ) (p +q ) . (3.12) l σf f l σf f l −σf 2 h | (pf + ql) The central part of the calculation are different loop-topologies with fermionic and scalar particles couplings to -even and -odd Higgs bosons. The CP µ1µ2 CP simplest topology is a triangle graph TΦ,p (q1, q2, mp), where q1, q2 denote momenta of the attached gluons. The shortcut Φ stands for all neutral Higgs bosons, HSM, h, H, A and mp is the mass of the scalar or fermionic loop- µ1µ2µ3 particle. The tensor BΦ,p (q1, q2, q3, mp) describes box-like contributions. The analytically and numerically most complex graphs are the pentagons µ1µ2µ3µ4 µ1µ2µ3µ4µ5 PΦ,p (q1, q2, q3, q4, mp) and hexagons HΦ,p (q1, q2, q3, q4, q5, mp). Charge-conjugation related diagrams, where the loop momentum is running clockwise and counter-clockwise, can be counted as one by exploiting Furry’s theorem [29]. This effectively reduces the number of diagrams by a factor of two. All briefly mentioned topologies differ in their tensor structure, depend- ing on, which particle couples to the Higgs boson, as well as on the Higgs boson coupling itself:

1) CP-odd Higgs boson A For the -odd Higgs boson A only couplings to fermions are taken CP into account. Massive squark loops can be safely neglected, because these contributions sum up to zero at amplitude level [8], due to the fact, that the both attached sfermions have to be in different bases (left-right basis L, R or mixed basis 1,2, see Eq. (A.25)) to give a non- vanishing contribution. Here the index f = t, b denotes top and bottom quark loops.

2) CP-even Higgs bosons HSM ,h,H The -even Standard Model Higgs boson H is produced only with CP SM 21 massive quark loops. The couplings of the light h and heavy H MSSM Higgs bosons differ by additional factors containing the mixing angle α of the neutral Higgs-doublet components and the mixing angle β of both vacuum expectation values. In the MSSM, the -even Higgs bosons can also couple to squarks. CP µ1µ2 However, a single triangle-tensor T h,H,f˜(q1, q2, mf˜) is proportional to an ǫ−1-pole in dimensional regularization (DREG) and, hence, it is UV- divergent. The MSSM provides also a further vertex, which is composed of two sfermions and two gluons (see Appendix D). Now, this ǫ−1- µ1µ2 pole cancels by addition of a two-point function S (q , q , m ˜) (see h,H,f˜ 1 2 f Fig. B.3 in App. B), containing theq ˜qgg˜ -vertex. The new divergence free triangle tensor is given by

µ1µ2 µ1µ2 µ1µ2 T (q , q , m ˜)= T (q , q , m ˜)+ S (q , q , m ˜) . (3.13) h,H,f˜ 1 2 f h,H,f˜ 1 2 f h,H,f˜ 1 2 f

All contributing topologies are described in more detail in Appendix B. Furthermore, polarization vectors of on-shell gluons with a triangle-loop in- sertion can be expressed by effective polarization vectors,

p,µ1 ǫµ2 (qi) µ1µ2 ei Φ = 2 TΦ,p (qi, qi + PΦ, mp) , (3.14) (qi + PΦ) where qi is the external gluon momentum, while PΦ denotes the momentum of the Higgs boson Φ. All coupling constants and loop factors are conveniently absorbed into an overall factor Fp. Detailed expressions for Fp are declared in the particular two or three jet process.

3.3 Higgs production in association with two jets

The production of the neutral Higgs bosons Φ in association with two jets, 4 at order αs, can be carried out via the sub-processes q q q q Φ , q Q q Q Φ , qg qg Φ , gg g g Φ . (3.15) → → → → The first two entries denote scattering of identical and non-identical quark flavors. The two last entries describe quark-gluon and gluon-gluon scatter- ing. Further contributions can be achieved by crossing relations using the

22 sign factors Si, introduced in Eq. (3.2) and the fact, that the Lagrangian is invariant under the application of the charge-conjugation operator Cˆ = γ0γ2,

quark-quark scattering: • q q q q Φ (via C invariance of amplitude) , −→ q q q q Φ , (3.16) −→ q q g g Φ , −→

quark-gluon scattering: • g q g q Φ (flipped beams) , −→ qg q g Φ (via C invariance of amplitude) , (3.17) −→ g q g q Φ (via C invariance of amplitude + flipped beams) , −→

gluon-gluon scattering: • g g q q Φ , −→ g g g g Φ . (3.18) −→

2j The overall factor Fp reads as

g4 F 2j = S = α2 (3.19) p Cp 16π2 Cp s The abbreviation stands for the Yukawa coupling to fermions and for the Cp Higgs coupling to sfermions. Detailed informations on couplings are given in

Appendix A. The index p = t, b, t˜1,2, ˜b1,2 denotes top and bottom quarks as well as stop and sbottom squarks.

3.3.1 qQ qQΦ and qq qqΦ → → The sub-process q Q q Q Φ of Eq. (3.15), depicted on the left side → in Fig. 3.2, is the simplest contribution to Φ + 2 jet production, if the Higgs boson couples to fermions only. Here Q denotes a different flavor. For identical quarks, an additional diagram can be obtained by interchang- ing the final-state-quarks. Higgs couplings to scalar quarks yield additional

23 Figure 3.2: Quark and squark contributions to qq qqΦ amplitudes. → contributions with squark-triangles. Furthermore, the MSSM provides new vertex-structures, which generate new loop-topologies, e.g. the two-point- loop, shown in the right panel of Fig. 3.2. At most, there are 6 diagrams, if both coupling types are taken into account. Following Ref. [2, 3, 30], the amplitude for different flavors and loop particles can be written as

qQ = F qQ,2jJ µ1 J µ2 T Φ,p (q , q , m ) ta ta A p 21 43 µ1µ2 1 2 p i2i1 i4i3 p X = qQ ta ta . (3.20) A2143 i2i1 i4i3 The overall factor

qQ,2j 0 0 0 0 2j Fp = S1 S2 S3 S4 4 p1 p2 p3 p4 Fp (3.21) q includes, in addition, normalization factors of external quark spinors (see Eq. (3.2) and (3.19)). External quark lines are expressed by the effective quark currents, introduced in Eq. (3.9). Both gluons attached to the tri- Φ,p angle tensor Tµ1µ2 (q1, q2, mp) are assumed as out-going and have momenta q = p p and q = p p . The sum of fermionic and sfermionic charge- 1 2 − 1 2 4 − 3 conjugated triangle graphs T µ1µ2 and sfermionic two-point function Sµ1µ2 are proportional to the single color factor δa2a3 (B.5). The two color factors δa1a2 and δa3a4 , coming from propagators of the virtual gluons, and the remain- a1 a4 ing color generators ti2i1 and ti4i3 of both gluons-vertices attached to the fermion currents of Eq. (3.9), yield the following simple color structure for the amplitude

a1 a1a2 a2a3 a3a4 a4 a a ti2i1 δ δ δ ti4i3 = ti2i1 ti4i3 . (3.22)

For identical fermions, Pauli-interference has to be considered additionally. That means, that to the amplitude of equation (3.20) a term with inter-

24 .

Figure 3.3: Quark and squark contributions to qg qgΦ amplitudes. → changed final state quarks must be added

qq = qq ta ta qq ta ta . (3.23) A A2143 i2i1 i4i3 −A4123 i4i1 i2i3 Fermi-statistics provides a relative sign-factor swap for the second term. Fi- nally, the squared amplitude, summed over initial and final particle colors, becomes N 2 1 qq 2 = 2 + 2 − +2 Re ∗ |A | |A2143| |A4123| 4 A2143A4123 color X  
N 2 1 − . (3.24) × 4N Setting = 0 in Eq. (3.24), one gets the squared matrix elements for A4123 qQ. A

3.3.2 qg qgΦ → This sub-process contains two (anti-)quarks and two gluons as external par- ticles. The momenta of the initial and final state gluons are set to q = q i − i and qf = +qf respectively. In the case of fermion-loops, there are only ten different Feynman graphs, if Furry’s theorem is taken into account. That are seven graphs with a triangle insertion and three box graphs with cyclic per- mutation of the attached gluons. They are sketched in the Fig. 3.3. The red crosses denote further positions for possible triangle-loop or scalar two-point

25 function insertions. In the sfermionic case, additional triangle graphs T µ1µ2µ3 B,Φ,f˜ with a box-like color structure and aq ˜qgg˜ -vertex (see Appendix B1.5) appear additionally to the 17 graphs composed of squark-triangles with accompany- ing squark two-point functions and squark-boxes. However, the new triangle contribution has a vanishing color factor and, hence, does not contribute to the sub-process (a brief proof of this statement is shown in Appendix B.1.5). The amplitude for the sub-process qg qgΦ can be written in a compact → form, using the notations of Chapter 3.2,

qg = qg ǫµ1 ǫµ2 = F qg,2j A Aµ1µ2 1 2 p p X ta1 ta2 2 (ep ) q , 1 + 2, q (ep ) 1 i2i1 1Φ σ1 2 1 2Φ σ1 × ( h | | i h | | i
h i a2 a1 p p + t t 2 (e )σ1 q1, 1 + 2, q2 (e )σ1 1 i2i1 h | 2Φ | i h | 1Φ | i
h i + ta1 , ta2 2 ep ǫ J q ep J ǫ (p p ) i2i1 1Φ 2 21 2 1Φ 21 2 2 1 " · · − · · −    ep q J ǫ 2 ep ǫ J q ep q J ǫ − 1Φ · 2 21 · 2 − 2Φ · 1 21 · 1 − 2Φ · 1 21 · 1   ep J ǫ (p p ) + J G,µ1 J µ2 T Φ,p (q + q ,p p , m ) − 2Φ · 21 1 · 2 − 1 12 21 µ1µ2 1 2 2 − 1 p  Φ,p µ1 µ3 µ3 Bµ1µ2µ3 (q1, q2,p2 p1, mp) ǫ1 ǫ1 J21 . (3.25) − − #)

Spinor normalization factors are absorbed again into the overall factor

F qg,2j = S S 2 p0 p0 δ F 2j . (3.26) p − 1 2 1 2 σ1σ2 p q The first four terms correspond to the Compton-like graphs with effective p,µ polarization vectors ei Φ (3.14). In the sfermionic case, one has to keep in mind, that the effective polarization vector is composed of two parts (see Eq. (3.13))

f,µ˜ 1 ǫµ2 (qi) µ1µ2 µ1µ2 e = T (q , q + P , m ˜)+ S (q , q + P , m ˜) . i Φ 2 Φ,f˜ i i Φ f Φ,f˜ i i Φ f (qi + P ) h i(3.27)

26 The color factor ta1 ta2 results from the color contribution δa2a3 of the i2i1 gluon propagator of the effective polarization vector and both gluon ver-
tices ta3 and ta1 . The same applies for ta2 ta1 . The remaining terms i2i1 i2i1 i2i1 correspond to three graphs containing a three-gluon-verte
x with either at- tached effective polarization vector and effective quark current or both ef- fective currents of Eqs. (3.9) and (3.10). Box diagrams enter via the tensor µ1µ2µ3 BΦ,p (q1, q2, q3, mp), which is described in more detail in Appendix B.2. These diagrams are proportional to the structure constant f abc with con- c tracted ti2i1 , coming from the effective quark current of Eq.(3.9). Finally, the amplitude can be separated into two parts with independent color structures

qg = ta1 ta2 qg + ta2 ta1 qg . (3.28) A i2i1 A12 i2i1 A21

The indices 12 and 21 label amplitudes with interchanged external gluons. Thus, the resulting color-summed squared amplitude takes the form,

2 N 2 1 qg 2 = qg 2 + qg 2 − 2 Re qg qg ∗ A A12 A21 4N − A12 A21 color
X   h
i N 2 1 − . (3.29) × 4N

3.3.3 gg ggΦ → Here, the momenta of gluons in initial and final state are set again to q = q i − i and qf = +qf respectively. Taking Furry’s theorem into consideration, 49 fermionic diagrams contribute to that sub-process: 19 graphs with triangle insertions, 18 box contributions and 12 pentagon diagrams. In the sfermionic case, additionally to the mentioned topologies, three triangle graphs with two q˜qgg˜ -vertices and 18 box diagrams with oneq ˜qgg˜ -vertex provide further con- tributions. Both new topologies have a pentagon-like color structure (more in appendix B). Here, it is strategically favorable to start with the pentagons and investigate their color structure. The four attached gluons give rise to 4! = 24 different color traces of the form,

tr tai taj tak tal , with i, j, k, l =1, 2, 3, 4 and i = j = k = l , 6 6 6 (3.30)  

27 from which (4 1)! = 6 are independent only due to the invariance property − of the trace under cyclic permutations. They can be combined to three independent real-valued color structures,

a1 a2 a3 a4 a1 a4 a3 a2 c1 = tr t t t t + tr t t t t ,

a1 a3 a4 a2 a1 a2 a4 a3 c2 = trt t t t  + trt t t t  , (3.31)

a1 a4 a2 a3 a1 a3 a2 a4 c3 = trt t t t  + trt t t t  .     All ci correspond to the sum of two pentagons with opposite loop momentum

flow. Finally, to every ci, four charge-conjugated pentagons can be assigned with cyclic permutation of external gluons. The evaluation of the color traces yields

1 2 c = δa1a2 δa3a4 + da1a2mda3a4m f a1a2mf a3a4m , 1 4 N −   1 2 c = δa1a3 δa4a2 + da1a3mda4a2m f a1a3mf a4a2m , (3.32) 2 4 N −   1 2 c = δa1dδa2a3 + da1a4mda2a3m f a1a4mf a2a3m , 3 4 N −   for which some SU(N)-identities of Appendix C were used. The color struc- ture of the remaining diagrams is proportional to two structure constants e.g. f a1a2mf ma3a4 . Furthermore, the combination of two structure constants can be expressed with help of the following identities by the color coefficients of Eq. (3.31)

1 c c = f a1a2mf a3dm = f a1a2mf a3dm =2 c c , 1 − 2 −2 ⇒ 2 − 1 1
c c = f a1dmf a2a3m = f a1dmf a2a3m =2 c c , (3.33) 3 − 1 −2 ⇒ 1 − 3 1
c c = f a1a3mf da2m = f a1a3mf da2m =2 c c . 2 − 3 −2 ⇒ 3 − 2
The sum of all differences of the ci satisfies the Jacobian identity,

2 c c + c c + c c − 1 − 2 3 − 1 2 − 3 h i = f a
1a2mf ma3a4 +
f a1a4mf ma
2a3 + f a1a3mf ma4a2 =0 . (3.34)

28 Figure 3.4: Quark- and squark-box contributions to gg ggΦ ampli- → tudes.

In terms of these color coefficients, the complete amplitude for gg gg Φ → can be decomposed into three separately gauge invariant sub-amplitudes,

3 3 gg gg = c gg = F 2j c ǫµ1 ǫµ1 ǫµ1 ǫµ1 , (3.35) A iAi p iAi,µ1µ2µ3µ4 1 2 3 4 i=1 i=1 X X 2j with the overall factor Fp from Eq. (3.19). Finally, all remaining diagrams with box, triangles and two-point topologies are explained briefly:

1) Fermionic and sfermionic box diagrams Box diagrams are attached to the gluon-current of Eq. (3.10), as shown in

Fig. 3.4. The momentum of the virtual gluon is denoted by qij = qi + qj. For a1a2m ma3a4 a gluon permutation (q12, q3, q4) with color factor f f , the partial amplitude can be written as,

Box =2 c c B α µ2 µ3 (q , q , q , m )J G ǫ ǫ . (3.36) A12,3,4 2 − 1 Φ,p 12 3 4 p 12, α 3, µ2 4, µ3
There are, of course, two further contributions with permutations (q3, q4, q12) and (q4, q12, q3) with the same color structure, which can be achieved via cyclic permutation of the attached gluons. Those are already taken into account in the definition of the box tensor in Eq. (B.28). For the remaining 15 diagrams, all color factors and momenta sets are listed here

(q , q , q ), (q , q , q ), (q , q , q ) = f a1a3mf ma2a4 = 2(c c ) , • 13 2 4 2 4 13 4 13 2 ⇒ 2 − 3 (q , q , q ), (q , q , q ), (q , q , q ) = f a1a4mf ma2a3 = 2(c c ) , • 14 2 3 2 3 14 3 14 2 ⇒ 1 − 3 (q , q , q ), (q , q , q ), (q , q , q ) = f a2a3mf ma1a4 = 2(c c ) , • 23 1 4 1 4 23 4 23 1 ⇒ 1 − 3 (q , q , q ), (q , q , q ), (q , q , q ) = f a2a4mf ma1a3 = 2(c c ) , • 24 1 3 1 3 24 3 24 1 ⇒ 2 − 3 29 Figure 3.5: Pentagon-like squark-box contribution to gg ggΦ am- → plitudes.

(q , q , q ), (q , q , q ), (q , q , q ) = f a3a4mf ma1a2 = 2(c c ) . • 34 1 2 1 2 34 2 34 1 ⇒ 2 − 1

2) Sfermionic box diagrams withq ˜qgg˜ -vertex The box diagram, depicted in Fig. 3.5, is contracted directly with four polar- ization vectors of the external gluons. Theq ˜qgg˜ -vertex provides a pentagon- like color structure, which can be expressed very easily by the color coeffi- cients of Eq. (3.31). Hence, for a gluon permutation (q12, q3, q4), the partial amplitude with the box tensor of Eq. (B.48) reads as

BoxP = tr ta1 ta2 ta3 ta4 + tr ta2 ta1 ta3 ta4 + tr ta4 ta3 ta1 ta2 A12,3,4   a4 a3 a2 a1  µ1µ2 µ3 µ4    + tr t t t t B (q , q , q , m ˜) ǫ 1 ǫ 2 ǫ 3 ǫ 4 P,Φ,f˜ 12 3 4 f 1 µ 2 µ 3 µ 4 µ  µ1 µ2µ3 µ4 =(c + c ) B (q , q , q , m ˜) ǫ 1 ǫ 2 ǫ 3 ǫ 4 . 1 2 P,Φ,f˜ 12 3 4 f 1, µ 2, µ 3, µ 4, µ (3.37)

The definition of Bµ1µ2µ3µ4 (q , q , q , m ) contains also two additional con- P,Φ,f˜ 12 3 4 f˜ tributions with cyclicly permuted momenta sets. The remaining 15 graphs with corresponding color factors are given by,

(q , q , q ), (q , q , q ), (q , q , q ) = (c + c ) , • 13 2 4 2 4 13 4 13 2 ⇒ 2 3 (q , q , q ), (q , q , q ), (q , q , q ) = (c + c ) , • 14 2 3 2 3 14 3 14 2 ⇒ 1 3 (q , q , q ), (q , q , q ), (q , q , q ) = (c + c ) , • 23 1 4 1 4 23 4 23 1 ⇒ 1 3 (q , q , q ), (q , q , q ), (q , q , q ) = (c + c ) , • 24 1 3 1 3 24 3 24 1 ⇒ 2 3 (q , q , q ), (q , q , q ), (q , q , q ) = (c + c ) . • 34 1 2 1 2 34 2 34 1 ⇒ 1 2 30 Figure 3.6: Quark- and squark-triangle contributions to gg ggΦ → amplitudes.

3) Fermionic and sfermionic triangle diagrams All subsequent described contributions with triangle-loop insertions are il- lustrated in Fig. 3.6. The first three contributions can be built up with the µ1µ1 general triangle tensor TΦ,p (q1, q2, mp) and two gluon currents of Eq (3.10). a1a2m For a momentum set, (q12, q34), the partial amplitude with color factor f f ma3a4 is given by,

Tri 1 =2(c c ) T Φ,p (q , q , m ) J G,µ1 J G,µ2 . (3.38) A12,34 2 − 1 µ1µ2 12 34 p 12 34 The color factors for the remaining two gluon momenta sets are,

(q , q ) = f a1a3mf a2a4m = 2(c c ) , • 13 24 ⇒ 2 − 3 (q , q ) = f a1a4mf a2a3m = 2(c c ) . • 14 23 ⇒ 1 − 3 For the next 15 diagrams, the building blocks are a three-gluon vertex, the gluon current of Eq. (3.10) and an effective polarization vector of Eq. (3.14). p a1a2m ma3a4 For a gluon momenta configuration (q12, e3Φ, q4) with color factor f f , the partial amplitude can be written as,

Tri 2 =2(c c ) ep ǫ (q + P q ) J G + ep J G (q q A12,3,4 2 − 1 3Φ · 4 3 Φ − 4 · 12 3Φ · 12 12 − 3 h P ) ǫ + ǫ J G (q + q ) ep , q ) , (3.39) − Φ · 4 4 · 12 12 4 · 3Φ 4 i where PΦ denotes the Higgs momentum. The color coefficients for the other 11 diagrams are

31 (q , q , ep ) = f a1a2mf a3a4m =2 c c , • 12 3 4Φ ⇒ 2 − 1 (q , ep , q ) = f a1a3mf a2a4m =2c c
, • 13 2Φ 4 ⇒ 2 − 3 (q , q , ep ) = f a1a3mf a2a4m =2c c
, • 13 2 4Φ ⇒ 2 − 3 (q , ep , q ) = f a1a4mf a2a3m =2c c
, • 14 2Φ 3 ⇒ 1 − 3 (q , q , ep ) = f a1a4mf a2a3m =2c c
, • 14 2 4Φ ⇒ 1 − 3 (q , ep , q ) = f a2a3mf a1a4m =2c c
, • 23 1Φ 4 ⇒ 1 − 3 (q , q , ep ) = f a2a3mf a1a4m =2c c
, • 23 1 4Φ ⇒ 1 − 3 (q , ep , q ) = f a2a4mf a1a3m =2c c
, • 24 1Φ 3 ⇒ 2 − 3 (q , q , ep ) = f a2a4mf a1a3m =2c c
, • 24 1 3Φ ⇒ 2 − 3 (q , ep , q ) = f a3a4mf a1a2m =2c c
, • 34 1Φ 2 ⇒ 2 − 1 (q , q , ep ) = f a3a4mf a1a2m =2c c
. • 34 1 2Φ ⇒ 2 − 1 The four-gluon (Appendix D) vertex gives rise
to four additional contribu- tions with one attached effective polarization vector. In addition, this vertex is composed of three terms, which are separately proportional to two struc- p ture constants. For a permutation (e1Φ, ǫ2, ǫ3, ǫ4), the partial amplitude reads as follows

Tri 3 =2 c c ep ǫ ǫ ǫ ep ǫ ǫ ǫ A1,2,3,4 2 − 1 1Φ · 3 2 · 4 − 1Φ · 4 2 · 3 h i +2 c c
ep ǫ
ǫ ǫ
ep ǫ
ǫ ǫ
2 − 3 1Φ · 2 3 · 4 − 1Φ · 4 2 · 3 h i +2 c c
ep ǫ
ǫ ǫ
ep ǫ
ǫ ǫ
. (3.40) 1 − 3 1Φ · 2 3 · 4 − 1Φ · 3 2 · 4 h i The remaining permutations
with the
same
color structure
are

(ǫ , ep , ǫ , ǫ ), (ǫ , ǫ , ep , ǫ ), (ǫ , ǫ , ǫ , ep ) . • 1 2Φ 3 4 1 2 3Φ 4 1 2 3 4Φ

4) Sfermionic triangle diagrams with twoq ˜qgg˜ -vertices Finally, the last triangle topology (B.1.6), depicted in Fig. 3.7, is contracted directly with four polarization vectors of the external gluons. The twoq ˜qgg˜ - vertices provide a pentagon-like color structure, which can be also expressed

32 Figure 3.7: Pentagon-like squark-triangle contributions to gg ggΦ → amplitudes. very easily by the color coefficients of Eq. (3.31). Hence, for a gluon permu- tation (q12, q34) the partial amplitude with the triangle tensor of Eq. (B.1.6) is given by,

TriP =2 tr ta1 ta2 ta3 ta4 + tr ta1 ta2 ta4 ta3 + tr ta2 ta1 ta3 ta4 A12,34  a2a1 a4 a3 P,Φ,f˜   µ1 µ2 µ3 µ4  + tr t t t t Tµ1µ2µ3µ4 (q12, q34, mf˜) ǫ1 ǫ2 ǫ3 ǫ4

 P,Φ,f˜ µ1 µ2 µ3 µ4 =2(c1 + c2) Tµ1µ2µ3µ4 (q12, q34, mf˜) ǫ1 ǫ2 ǫ3 ǫ4 . (3.41)

The other permutations are

(q , q ) = 2 tr ta1 ta3 ta2 ta4 + tr ta1 ta3 ta4 ta2 + tr ta3 ta1 ta2 ta4 • 13 24 ⇒        a3 a1 a4 a2 +tr t t t t =2(c2 + c3) ,   (q , q ) = 2 tr ta1 ta4 ta2 ta3 + tr ta1 ta4 ta3 ta2 + tr ta4 ta1 ta2 ta3 • 14 23 ⇒        a4 a1 a3 a2 +tr t t t t =2(c1 + c3) .   The sum over colored gluons of the squared amplitude becomes

3 gg 2 = gg gg ∗ c c |A | Ai Aj i j color i,j=1 col X X
X 3 3 = gg 2 + gg gg ∗ (3.42) C1 |Ai | C2 Ai Aj i=1 i,j=1; i6=j X X

33 with N 2 1 N 4 2N 2 +6 c c = − − , (no sum. over i), (3.43) C1 ≡ i i 8N 2 col

X N 2 1 3 N 2 c c = − − , i = j . (3.44) C2 ≡ i j 4N 2 6 col

X 3.4 Higgs production in association with three jets

For the production of Higgs bosons Φ in a 2 4 QCD process of the order 5 → αs, the former sub-processes are complemented by an additional gluon in the final-state with momentum qf = +qf . The production of neutral Higgs bosons Φ can then be carried out via the following sub-processes,

q q qqg Φ, q Q qQg Φ, qg qgg Φ, gg ggg Φ. (3.45) → → → → Further crossing related sub-processes are given by,

quark-quark scattering: • q q q q g Φ (via C invariance of amplitude) , −→ q q q q g Φ , (3.46) −→ q q Q Qg Φ , −→ q q ggg Φ , −→ quark-gluon scattering: • q g q q q Φ , −→ q g Q Q q Φ , −→ g q q q q Φ (flipped beams) , (3.47) −→ g q gqg Φ (flipped beams) , −→ q g qgg Φ (via C invariance of amplitude) , −→ g q g q g Φ (via C invariance of amp. + flipped beams) , −→

34 gluon-gluon scattering: • g g q q g Φ , −→ g g ggg Φ . (3.48) −→

3j Here, the overall factor Fp , containing all coupling constants, reads as

g2 5/2 F 3j = S = α5 (3.49) p Cp 4π Cp s   The present calculation does not include contributions with squarks. To shorten expressions for amplitudes, one can define new quark currents con- taining the emission of a gluon close to an external quark in the initial or final state 1 1 J µ ( q )= δ χ† (¯p )(σµ) (p q ) (ǫ ) χ (¯p ) fi ∗ l σf σi σf f τ i − l −σi l σi σi i (p q )2 q2 i − l fi 1 = δ f (σµ) q , l , (3.50) σf σi h | τ | i i(p p + q )2 i − f l 1 1 J µ (q )= δ χ† (¯p )(ǫ ) (p +q ) (σµ) χ (¯p ) fi l σf σi σf f l σf f l −σf 2 τ σi i 2 ∗ (pf + ql) qfi 1 = δ f, q (σµ) i . (3.51) σf σi h l| τ | i(p p + q )2 i − f l Gluon radiation in initial or final state is denoted by a star. It tags the position of gluon emission. Further fermion currents with the effective po- larization vector of Eq. (3.14) can be defined in a similar way

J µ ( q , Φ) fi ∗ l 1 1 = δ χ† (¯p )(σµ) (p +q + P ) (ep ) χ (¯p ) σf σi σf f τ i l Φ −σi l Φ σi σi i 2 2 (pi + ql + PΦ) qfi 1 = δ f (σµ) q , i, Φ , (3.52) σf σi h | τ | l i(p p + q + P )2 i − f l Φ J µ (q , Φ) fi l∗ 1 1 = δ χ† (¯p )(ep ) (p q P ) (σµ) χ (¯p ) σf σi σf f l Φ σf f − l − Φ −σf (p q P )2 τ σi i q2 f − l − Φ fi 1 = δ f, q , Φ (σµ) i . (3.53) σf σi h l | τ | i(p p + q + P )2 i − f l Φ 35 This formalism also allows for further definitions of quark currents, but with two emitted gluons 1 J µ (q , q )= δ χ† (¯p )(ǫ ) (p +q ) (σµ) fi l m σf σi σf f l σf f l −σf 2 τ (pf + ql) 1 1 (p q ) (ǫ ) χ (¯p ) × i − m −σi m σi σi i (p q )2 q2 i − m fi 1 = δ f, q (σµ) i, q . (3.54) σf σi h l| τ | mi(p p + q + q )2 i − f l m Further currents, containing an attached Higgs boson, are

J µ (q , q , Φ) = δ χ† (¯p )(ep ) (p q P ) fi l∗ m σf σi σf f l Φ σf f − l − Φ −σf 1 1 1 (σµ) (p +q ) (ǫ ) χ (¯p ) × (p q P )2 τ i m −σi m σi σi i (p + q )2 q2 f − l − Φ i m fi 1 = δ f, q , Φ (σµ) i, q , (3.55) σf σi h l | τ | mi(p p + q + q + P )2 i − f l m Φ 1 J µ (q , q , Φ) = δ χ† (¯p )(ǫ ) (p q ) (σµ) fi l ∗ m σf σi σf f l σf f − l −σf (p q )2 τ f − l 1 1   p (pi + qm + PΦ)−σi (em Φ)σi χσi (¯pi) 2 2 × (pi + qm + PΦ) qfi 1 = δ f, q (σµ) i, q , Φ . (3.56) σf σi h l| τ | m i(p p + q + q + P )2 i − f l m Φ Finally, due to a large number of diagrams in the subsequent sub-processes, it is useful to introduce a shorthand notation for the three-gluon vertex without coupling constant

GVµ1µ2µ3 (k,p,q)= gµ1µ2 (k p)µ3 + gµ2µ3 (p q)µ1 + gµ3µ(q k)µ2 , 3 − − − h (3.57)i with color coefficient f a1a2a3 . The four-gluon vertex is composed of three parts with different color and tensor structures (App. D), which are treated by MadGraph as individual diagrams. For the illustration of partial ampli- tudes, the first part of the vertex is used only

GVµ1µ2µ3µ4 = (gµ1µ3 gµ2µ4 gµ1µ4 gµ2µ3 ) , (3.58) 4 − − where f a1a2mf ma3a4 is the corresponding color factor.

36 1, 2, 3, 4 5, 6, 7, 8 9, 10, 11 12, 13, 14

Figure 3.8: Quark-loop contributions to qq qqgΦ amplitudes. →

3.4.1 qQ qQgΦ and qq qqgΦ → → The simplest contribution contains four quarks pairwise of different flavor and one gluon as external particles. In total, there are 14 diagrams, in which 11 have a triangle insertion and the remaining 3 a box topology. These diagrams are illustrated in Fig. 3.8, where different configurations are denoted by a red cross. Partial amplitudes with the emission of a gluon from a quark line from initial and final state are listened here

qQ,Tri 1 = ta2 ta1 ta2 J µ1 ( q) J µ2 T Φ (p p + q,p p , m ), A∗21,43 21 43 21 ∗ 43 µ1µ2 2 − 1 4 − 3 t (3.59)
qQ,Tri 2 = ta1 ta2 ta2 J µ1 (q ) J µ2 T Φ (p p + q,p p , m ), A21∗,43 21 43 21 ∗ 43 µ1µ2 2 − 1 4 − 3 t (3.60)
qQ,Tri 3 = ta2 ta2 ta1 J µ1 J µ2 ( q) T Φ (p p ,p p + q, m ), A21,∗43 21 43 21 43 ∗ µ1µ2 2 − 1 4 − 3 t (3.61)
qQ,Tri 4 = ta2 ta1 ta2 J µ1 J µ2 (q ) T Φ (p p ,p p + q, m ). A21,43∗ 21 43 21 43 ∗ µ1µ2 2 − 1 4 − 3 t (3.62)
These four contributions provide the basis for the color structure of this sub- process. Hence, all subsequent partial amplitudes are distributed to that basis. Four similar amplitudes with the same color factors can be constructed by removing the triangle insertion between both quark lines and replacing the usual polarization vector by an effective one

qQ,Tri 5 = ta2 ta1 ta2 J ( q, Φ) J , (3.63) A∗21,43 21 43 21 ∗ · 43 qQ,Tri 6 = ta1 ta2
ta2 J (q , Φ) J , (3.64) A21∗,43 21 43 21 ∗ · 43
37 qQ,Tri 7 = ta2 ta2 ta1 J J ( q, Φ) , (3.65) A21,∗43 21 43 21 · 43 ∗ qQ,Tri 8 = ta2 ta1 ta2
J J (q , Φ) . (3.66) A21,43∗ 21 43 21 · 43 ∗ The next two graphs contain
a three-gluon vertex and are given by

qQ,Tri 10 a2 a2 a1 a1 a2 µ1 Φ = t t t t t J T (p2 p1,p4 p3 + q, mt) A21,43 i2i1 − i2i1 21 µ1µ2 − − 2 Jµ2 p p ǫ(q)+ J ǫ(q) p p + ǫ(q) µ2 × 43 4 − 3 · 43 · 3 − 4 h + ǫ(q)µ2 p
p 2 q J ,
(3.67) 3 − 4 − · 43 qQ,Tri 9 a2 a2 a1 a1 a2
µ1 Φi = t t t t t J T (p2 p1,p4 p3 + q, mt) A21,43 i2i1 − i2i1 21 µ1µ2 − − 2 Jµ2 p p ǫ(q)+ J ǫ(q) p p + ǫ(q) µ2 × 43 4 − 3 · 43 · 3 − 4 h + ǫ(q)µ2 p
p 2 q J .
(3.68) 3 − 4 − · 43 i The remaining contribution with a triangle
insertion is composed of a three- gluon vertex, effective polarization vector of Eq. (3.14) and two quark cur- rents, defined in Eq. (3.9)

qQ,Tri 11 a2 a2 a1 a1 a2 p = t t t t t J21 J43 p1 + p4 p2 p3 e A21,43 i2i1 − i2i1 · − − · Φ + J ep p p + q + P J + J ep p p
43 · Φ 3 − 4 Φ · 21 21 · Φ 2 − 1 q P J .
(3.69) − − Φ · 43 In addition, one can insert
a box-topology into the diagram with a three-gluon vertex in three different ways. The 3! permutations of the attached gluons are reduced by Furry’s theorem to three graphs, which are already captured µ1µ2µ3 by the box tensor BΦ (q1, q2, q3, mt). The partial amplitude reads as

qQ,Box a3 a2 a2 a3 Φ µ1 = t t t t B (q,p2 p1,p4 p3, mt) ǫ(q) A21,43 − i2i1 µ1µ2µ3 − − J µ2 J µ3 .  (3.70) × 21 34 The color coefficient of the last four partial amplitudes is proportional to a structure constant f. It is useful to decompose it into the color basis of Eqs. (3.59-3.62)

f a1a2a3 ta2 ta3 = ita2 ta2 ta1 ta1 ta2 . (3.71) i2i1 i4i3 i2i1 − i4i3   38 Identical quark-flavors double the amount of diagrams by interchanging final states. This provides four additional color factors,

a2 a1 a2 a1 a2 a2 a2 a2 a1 a2 a1 a2 t t 41t23, t t 41t23, t41 t t 23, t41 t t 23 . (3.72) One has to
keep Pauli-interference
in mind,
that changes
the sign of the corresponding diagrams. In total, there are eight different color structures, which interfere among each other. All interference terms form a symmetric 8 8 matrix qq, containing products of Casimir-operators of the funda- × RGBij mental and adjoint representation of the SU(N) algebra. For N = 3, it is given by

24 3 6 21 8 1 10 1 − − −   3 24 21 6 1 10 1 8 − − −       6 21 24 3 10 1 8 1  − − −       21 6 3 24 1 8 1 10  qq 1  − − −  ij =   . (3.73) RGB 9    8 1 10 1 24 3 6 21  − − −       1 10 1 8 3 24 21 6  − − −       10 1 8 1 6 21 24 3  − − −       1 8 1 10 21 6 3 24   − − −    The squared amplitude, summed over initial- and final-particle color, be- comes 8 2 ∗ qq 2 = F qQ,3j qq Re , (3.74) |A | p RGBij Ai Aj color i,j=1 X   X h
i with

qQ,3j 0 0 0 0 3j Fp = S1 S2 S3 S4 4 p1 p2 p3 p4 Ft . (3.75) q For a sub-process with different quark-flavors, only the symmetric 4 4 part × of the qq matrix with i, j = 1,..., 4 should be used to calculate the RGBij squared expression of the amplitude.

39 1a 1b 1c

Figure 3.9: Quark-triangle contributions to qg qggΦ amplitudes. →

3.4.2 qg qggΦ → With MadGraph [31], 84 distinct diagrams were generated for the sub-process qg qggΦ within the effective theory. The correspondence between the full → and the effective theory is described in more detail in appendix E. Further- more, the application of abbreviations introduced in previous chapters allows an indexing of different diagrams according to certain features. This proce- dure was already used in the Higgs + 2 jets process. It ensures a better view over the large variety and number of diagrams. In the following, expressions for partial amplitudes for a fixed permutation of external particle momenta are shown only:

1) Triangle diagrams with effective polarization vector only Fig. 3.9 shows three different possibilities to attach a triangle loop to the emitted gluons

qg,Tri 1a a1 a2 a3 = t t t J21(q3 , q1, Φ) ǫ2 , (3.76) A21 i2i1 ∗ · qg,Tri 1b a1 a2 a3 p = t t t
J21(q3, q1) e , (3.77) A21 i2i1 · 2Φ qg,Tri 1c a1 a2 a3 = t t t
J21(q3, q1 , Φ) ǫ2 . (3.78) A21 i2i1 ∗ · There are in total 18 diagrams:
3! = 6 different permutations of the attached gluons and three possibilities of Higgs emission. The remaining color factors are

ta3 ta1 ta2 , ta1 ta3 ta2 , ta2 ta1 ta3 , i2i1 i2i1 i2i1 ta2 ta3 ta1
, ta3 ta2 ta1
.
(3.79) i2i1 i2i1 These six color structures
provide
the color basis for that sub-process.

40 2a 2b 2c

Figure 3.10: Quark-triangle contributions to qg qggΦ amplitudes. →

2) Triangle diagrams with one three-gluon vertex There are 24 different contributions, containing an effective polarization vec- tor or a triangle loop with off-shell gluons. Three possibilities, shown in Fig. 3.10, are available to construct the partial amplitudes with given re- sources

qg,Tri 2a = if a2a1m tmta3 J ( q , Φ) J G (q + q )2 , (3.80) A 21 21 ∗ 3 · 12 1 2 qg,Tri 2b = if a2a1mtmta3
J ( q ) J G (q , q , Φ) A 21 21 ∗ 3 · 12 1∗ 2 (q + q + P )
2 , (3.81) × 1 2 Φ qg,Tri 2c = if a2a1m tmta3 T µ1µ2 (p p + q , q + q , m ) A 21 Φ 2 − 1 3 1 2 t J µ1 ( q ) J G .
(3.82) × 21 ∗ 3 12 All appearing color factors can be reduced to the basis, given by Eqs. (3.76- 3.79)

if a2a1m tmta3 = ta2 ta1 ta3 ta1 ta2 ta3 , (3.83) i2i1 − i2i1 if a3a1mtmtab
= ta3 ta1 ta2 ta1 ta3 ta2
, (3.84) i2i1 − i2i1 if a2a3mtmta1
= ta2 ta3 ta1 ta3 ta2 ta1
, (3.85) i2i1 − i2i1 if a2a1mta3 tm
= ta3 ta2 ta1 ta3 ta1 ta2
, (3.86) i2i1 − i2i1 if a3a1mta2 tm
= ta2 ta3 ta1 ta2 ta1 ta3
, (3.87) i2i1 − i2i1 if a2a3mta1 tm
= ta1 ta2 ta3 ta1 ta2 ta3
. (3.88) i2i1 − i2i1

41 3a 3b 3c

Figure 3.11: Quark-triangle contributions to qg qggΦ amplitudes. →

3) Triangle diagrams with two three-gluon vertices A total of 15 dia- grams with a triangle insertion connected to two three-gluon vertices makes contributions: 9 with effective polarization vector and 6 with a triangle tensor attached to two virtual gluons. Construction possibilities for partial ampli- tudes, depicted in Fig. 3.11, are given by,

qg,Tri 3a = f a2a1mf a3mntn GVµ1µ2µ3 ( q , q P ,p p ) A i2i1 3 − 3 − 12 − Φ 1 − 2 ǫ J G (q , q , Φ) J , (3.89) × 3,µ1 12,µ2 1 2∗ 21,µ3 qg,Tri 3b = f a2a1mf a3mntn GVµ2µ3µ4 ( q ,p p + q ,p p ) A i2i1 3 − 3 2 − 1 3 2 − 1 T Φ (q ,p p + q , m ) J G,µ1 J ǫ × µ1µ2 12 2 − 1 3 t 12 21,µ3 µ4,3 1 , (3.90) × (p p + q )2 2 − 1 3 qg,Tri 3c = f a2a1mf a3mntn GVµ1µ2µ3 ( q P , q ,p p ) A i2i1 3 − 3 − Φ − 12 1 − 2 e3,Φ J G J . (3.91) × p,µ1 12,µ2 21,µ3 Here, only three different color factors appear in the calculation, which can be reduced to the color basis using the identity of Eq. (C.10) in the following way

f a2a1mf a3mntn =( ta2 ta1 ta3 + ta1 ta2 ta3 + ta3 ta2 ta1 ta3 ta1 ta2 ) , i2i1 − − i2i1 (3.92) f a3a1mf a2mntn =( ta3 ta1 ta2 + ta1 ta3 ta2 + ta2 ta3 ta1 ta2 ta1 ta3 ) , i2i1 − − i2i1 (3.93) f a2a3mf ma1ntn =( ta2 ta3 ta1 + ta3 ta2 ta1 + ta1 ta2 ta3 ta1 ta3 ta2 ) . i2i1 − − i2i1 (3.94)

42 4a 4b

Figure 3.12: Quark-triangle contributions to qg qggΦ amplitudes. →

5a 5b 5c

Figure 3.13: Quark-box contributions to qg qggΦ amplitudes. →

4) Triangle diagrams with four gluon vertex MadGraph [31] generates 12 graphs, which correspond to 4 four-gluon ver- tices. The partial amplitudes, shown in Fig. 3.12, reads as

qg,Tri 4a = f a1a2mf ma3ntn GV eµ1,p ǫµ2 ǫµ3 J µ4 , (3.95) A i2i1 4, µ1µ2µ3µ4 3Φ 2 3 21 qg,Tri 4b = f a1a2mf ma3ntn GV ǫµ1 ǫµ2 ǫµ3 A i2i1 4, µ1µ2µ3µ4 1 2 3 T µ4µ5 (q ,p p , m )J . (3.96) × Φ 123 2 − 1 t 21,µ5 The color structure is the same as in Eqs. (3.92 -3.94), except for a relative sign, due to different permutations of the involved gluons.

5) Box diagrams Partial amplitudes with a box tensor, depicted in Fig. 3.13, can be con- structed in the following ways

qg,Box 5a a2a1m m a3 Φ = if t t B (q2, q1,p2 p1 + q3, mt) A i2i1 µ1µ2µ3 − ǫµ1 ǫµ2 Jµ3 (q
) , (3.97) × 2 1 21 3∗ qg,Box 5b = f a2a1mf a3mntn BΦ (q , q ,p p , m ) A i2i1 µ1µ2µ3 12 3 2 − 1 t 43 J G, µ1 ǫµ2 J µ3 , (3.98) × 32 1 21 qg,Box 5c = f a2a1mf a3mntn GV BΦ (q , q ,p p + q , m ) A i2i1 3, µ3µ4µ5 µ1µ2µ3 2 1 2 − 1 3 t ǫµ2 ǫµ1 ǫµ4 J µ5 . (3.99) × 1 2 1 21 The decomposition of the color factors in terms of the basis elements was already shown in Eqs. (3.83 -3.88) and (3.92 -3.94).

6) Pentagon diagrams In the effective theory there is only one pentagon, that corresponds to a four-gluon vertex with an additional Higgs boson emission. Furthermore, it features the same tensor and color structure of a four-gluon vertex. The full loop theory provides 12 pentagon diagrams, if Furry’s theorem is taken into account. The color structure does not correspond to a four-gluon vertex and therefore has to be treated separately

(a1, a2, a3,d), (a2, a3,d,a1), (a3,d,a1, a2), (d, a1, a2, a3) perm.: • (a ,d,a , a ), (d, a , a , a ), (a , a , a ,d), (a , a ,d,a )  1 3 2 3 2 1 3 2 1 2 1 3

a4 1 a1a2a3 1 a1 a2 a3 a3 a2 a1 c1 t = d δi2i1 + t t t + t t t , (3.100) i2i1 −2 N 2 i2i1

(a1, a3,d,a2), (a3,d,a2, a1), (d, a2, a1, a3), (a2, a1, a3,d) perm.: • (a , a ,d,a ), (a ,d,a , a ), (d, a , a , a ), (a , a , a ,d)  1 2 3 2 3 1 3 1 2 3 1 2

a4 1 a1a2a3 1 a3 a1 a2 a2 a1 a3 c2 t = d δi2i1 + t t t + t t t , (3.101) i2i1 −2 N 2 i2i1

(a1,d,a2, a3), (d, a2, a3, a1), (a2, a3, a1,d), (a3, a1,d,a2) perm.: • (a , a , a ,d), (a , a ,d,a ), (a ,d,a , a ), (d, a , a , a )  1 3 2 3 2 1 2 1 3 1 3 2

a4 1 a1a2a3 1 a2 a3 a1 a1 a3 a2 c3 t = d δi2i1 + t t t + t t t , (3.102) i2i1 −2 N 2 i2i1
where the ci of Eq. (3.32) and SU(N)-identities of Appendix C were used. a4 The color generator ti2i1 comes from the attached quark current of Eq. (3.9) and is contracted with the free index a4 inside the ci. An example of such a diagram is shown in Fig. 3.14. In comparison to the effective theory, the

44 Figure 3.14: Quark-pentagon contributions to qg qggΦ amplitudes. → existence of the total symmetric structure constant da1a2a3 extends the color space by an additional dimension. Using the basis (3.76-3.79) at first, one can build up a symmetric 6 6 matrix qg, which contains again SU(N) × RGBij Casimir invariants

64 1 8 8 1 10 − −   1 64 8 10 1 8  − −       8 8 64 1 10 1  qg 1 − −  ij =   . (3.103) RGB 9    8 10 1 64 8 1  − −       1 1 10 8 64 8  − −       10 8 1 1 8 64   − −    The squared expression and interference term of the total symmetric struc- ture constant d with the color basis read as 1 2 10 C = da1a2a3 δ N==3 , (3.104) 1 − 2 N i2i1 36   1 † N=3 10 C = da1a2a3 δ (tai taj tak ) = , 2 −2 N i2i1 i2i1 72 h i i, j, k =1, 2, 3 and i = j = k . 6 6 For the loop-induced theory, the squared amplitude, summed over initial- and final-particle color, becomes

6 2 qq 2 = F qg,3j qg Re ∗ + C pen 2 |A | p RGBij Ai Aj 1 A color ( i,j=1 X   X h
i

45 6 ∗ +2 C Re pen , (3.105) 2 Ai A i=1 ) X  
i with F qg,3j = S S 2 p0 p0 δ F 3j . (3.106) p − 1 2 1 2 σ1σ2 p The shorthand ( pen) denotesq the sum of the 12 pentagon contributions. To A switch to the effective theory, one has to set both color coefficients Ci = 0 and take notice of the different color structure of the effective pentagons.

3.4.3 gg gggΦ → Using MadGraph [31], 380 diagrams were generated in the effective limit ap- proach. The full theory provides further hexagon-like topologies additionally to the already complicated pentagons. This circumstances make that pro- cess numerically very challenging. Due to the large number of diagrams and the length of the result, expressions for the amplitudes are not written here explicitly. It is strategically favorable to start with the hexagons and inves- tigate their color structure. The five external gluons give rise to 5! = 120 hexagons. They are proportional to 120 different color traces of the form tr tai taj tak tal tam with i, j, k, l, m =1,..., 5   and i = j = k = l = m , (3.107) 6 6 6 6 in which (5 1)! = 24 are independent, only due to the invariance property − of the trace under cyclic permutations. They are given explicitly by (1) tr ta1 ta2 ta3 ta4 ta5 , (2) tr ta1 ta2 ta3 ta5 ta4 , (3) tr ta1 ta2 ta4 ta5 ta3 , (4) trta1 ta2 ta5 ta4 ta3 , (5) trta1 ta3 ta4 ta5 ta2 , (6) trta1 ta3 ta5 ta4 ta2 , (7) trta1 ta4 ta5 ta3 ta2 , (8) trta1 ta5 ta4 ta3 ta2 , (9) trta1 ta3 ta4 ta2 ta5 , (10) tr ta1 ta3 ta2 ta5 ta4 , (11) tr ta1 ta3 ta3 ta2 ta4 , (12) tr ta1 ta4 ta2 ta5ta3 , (13) trta1 ta4 ta5 ta2 ta3 , (14) trta1 ta5 ta2 ta4 ta3 , (15) trta1 ta3 ta2 ta4 ta5 , (16) trta1 ta5 ta4 ta2 ta3 , (17) trta1 ta2 ta4 ta3 ta5 , (18) trta1 ta2 ta5 ta3 ta4 , (19) trta1 ta4 ta3 ta5 ta2 , (20) trta1 ta5 ta3 ta4 ta2, (21) trta1 ta4 ta3 ta2 ta5, (22) trta1 ta5 ta2 ta3 ta4 , (23) trta1 ta4 ta2 ta3 ta5 , (24) trta1 ta5 ta3 ta2 ta4  . (3.108)      

46 These 24 traces provide the color basis for the sub-process. The number of hexagons can be now reduced via Furry’s theorem [29] from 120 to 60. All charge-conjugated hexagons are then distributed to 12 differences of two op- posite color traces, which are built up from the color basis (3.108). In the previous sub-process, all pentagons were attached to the quark current of Eq. (3.9), which is here replaced by the gluon current (3.10). This replace- ment gives rise to 120 contributions with pentagon insertions. Using the color coefficients (3.31) and the commutator relation ta1 , ta2 = if a1a2a3 ta3 , one arrives at 10 basic momenta configurations for the pentagon graphs. The   G first momenta configuration 1, 2, 3, [4, 5] with gluon current J45 yields  c f ma4a5 = i tr ta1 ta2 ta3 ta4 ta5 tr tr ta1 ta2 ta3 ta5 ta4 1 − −  + trta1 ta4 ta5 ta3 ta2  tr ta1 ta5 ta4 ta3 ta2  , (3.109) −  c f ma4a5 = i trta1 ta3 ta4 ta5 ta2  trta1 ta3 ta5 ta4 ta2  2 − −  + trta1 ta2 ta4 ta5 ta3  trta1 ta2 ta5 ta4 ta3  , (3.110) −  c f ma4a5 = i trta1 ta4 ta5 ta2 ta3  trta1 ta5 ta4 ta2 ta3  3 − −  + trta1 ta3 ta2 ta4 ta5  trta1 ta3 ta2 ta5 ta4  . (3.111) −  The structure constantsf are contracted via the index m with the free index a4, replaced by m inside the ci. One has to keep in mind, that every ci is proportional to a sum of four charge-conjugated pentagons with cyclicly permuted gluons. The remaining permutations are listened in Appendix F.1. Graphs with triangle and box insertions are all proportional to a color factor composed of three structure constants f. It can be decomposed in terms of the color basis in the following way using the SU(N) identities of Appendix C

f a1a2mf ma3nf na4a5

= 2 i tr ta1 ta2 ta4 ta5 ta3 + tr ta1 ta2 ta5 ta4 ta3 tr ta1 ta2 ta3 ta4 ta5 − −  + trta1 ta2 ta3 ta5 ta4  trta2 ta1 ta4 ta5 ta3  + trta2 ta1 ta5 ta4 ta3  − trta2 ta1 ta3 ta4 ta5  + trta2 ta1 ta3 ta5 ta4  .  (3.112) −     47 All remaining Feynman graphs, containing a triangle- or a box-loop, are substantially similar to those shown in the former sub-process, but with quark-currents replaced by the corresponding gluon-currents. The rewriting rules of the Appendix E for triangle- and box-diagrams provide a simple switch to the full theory. Even the pentagons can be implemented in a quite easy way, because their color structure (see Eq. (3.109) and App. F.1) can be reduced directly to the color basis of Eq. (3.108). Finally a 24 × 24 symmetric color matrix gg, built up from the color basis, mixes all RGBij partial amplitudes and yields the squared expression of the sum of all partial amplitudes

24 2 ∗ qq 2 = F 3j gg Re . (3.113) |A | p RGBij Ai Aj color i,j=1 X   X h
i The entries of the matrix gg can be extracted directly from the corre- RGBij sponding sub-process, generated by MadGraph [31].

48 Chapter 4 Numerical implementation and checks

Analytic expressions for the amplitudes of the previous chapters were imple- mented in the Fortran program VBFNLO [4]. The tensor reduction of the loop contributions up to boxes is performed via Passarino-Veltman reduc- tion [32, 33], while for the pentagons the Denner-Dittmaier algorithm [34, 35] is used, which avoids the inversion of small Gram determinants emerging in planar configurations of the Higgs and the two final state partons. The program was numerically tested in several ways. Besides usual gauge- and Lorentz-invariance tests, the different topologies were also checked separately. µ1µ2 The contraction of a fermionic triangle-tensor TΦ,f (q1, q2, mf ) with gluon momentum qµ vanishes in the case of the -odd Higgs-boson, due to total i CP antisymmetry of the Levi-Civita symbol and in the -even case, since both CP form factors FL and FT , defined in Eqs. (B.8, B.9), are transversal. Hence, the Ward-identity for triangle loops is given by

µ1 Φ,f µ1 Φ,f q1 Tµ1µ2 (q1, q2, mf )= q2 Tµ1µ2 (q1, q2, mf )=0 . (4.1)

Contracting with external gluon momenta, the tensor expressions of fermionic boxes, pentagons and hexagons reduce to differences of triangles, boxes and pentagons respectively. With the tensor integrals as defined in the Ap- pendix B, the Ward-identities for the boxes read

qµ1 BΦ,f (q , q , q , m )= T Φ,f (q , q , m ) T Φ,f (q , q , m ) , (4.2) 1 µ1µ2µ3 1 2 3 f µ2µ3 12 3 f − µ2µ3 2 3 f qµ2 BΦ,f (q , q , q , m )= T Φ,f (q , q , m ) T Φ,f (q , q , m ) , (4.3) 2 µ1µ2µ3 1 2 3 f µ1µ3 1 23 f − µ1µ3 12 3 f qµ3 BΦ,f (q , q , q , m )= T Φ,f (q , q , m ) T Φ,f (q , q , m ) , (4.4) 3 µ1µ2µ3 1 2 3 f µ1µ2 1 2 f − µ1µ2 1 23 f

49 where the abbreviation qij = qi + qj has been used. Similarly, for the pen- tagons one finds

µ1 Φ,f q1 Pµ1µ2µ3µ4 (q1, q2, q3, q4, mf ) = BΦ,f (q , q , q , m ) BΦ,f (q , q , q , m ) , (4.5) µ2µ3µ4 12 3 4 f − µ2µ3µ4 2 3 4 f µ2 Φ,f q2 Pµ1µ2µ3µ4 (q1, q2, q3, q4, mf ) = BΦ,f (q , q , q , m ) BΦ,f (q , q , q , m ) , (4.6) µ1µ3µ4 1 23 4 f − µ1µ3µ4 12 3 4 f µ3 Φ,f q3 Pµ1µ2µ3µ4 (q1, q2, q3, q4, mf ) = BΦ,f (q , q , q , m ) BΦ,f (q , q , q , m ) , (4.7) µ1µ2µ4 1 2 34 f − µ1µ2µ4 1 23 4 f µ4 Φ,f q4 Pµ1µ2µ3µ4 (q1, q2, q3, q4, mf ) = BΦ,f (q , q , q , m ) BΦ,f (q , q , q , m ) . (4.8) µ1µ2µ3 1 2 3 f − µ1µ2µ3 1 2 34 f In the exactly same manner, one can continue with this procedure for the hexagons [36]. These identities were tested numerically and they, typically, are satisfied at least at the 10−9 level, when using Denner-Dittmaier reduc- tion for the tensor integrals. With decreasing number of external legs the accuracy of these identities improves. In addition, one can perform a QED-check for the pentagons. Replacing gluons by photons and considering the process γγ γγΦ, diagrams with → three- and four-gluon-vertices vanish, because these structures are not avail- able in an Abelian theory. The amplitude is simply given by the sum of all pentagon graphs, without color factors. When contracting with an external gauge boson momentum, one obtains zero, since boxes are not allowed for photons, by Furry’s theorem. The amplitudes pass this test as well. Squared expressions of amplitudes with squark-contributions and the Higgs couplings to up-type and down-type squarks were checked numerically for a selection of randomly generated phase space points against the FeynArts, FormCalc [37, 38, 39] package and agreed at least at 10−6 level. Also full cross sections of individual sub-processes in the Φjj production coincided within the integration error, which was below 1%. To check the full scattering amplitudes, one can make use of the heavy-top effective Lagrangian, which is explained in more detail in App. E. As mt becomes large, the results calculated with full fermion loops must approach the approximate ones, derived from the effective Lagrangian. This check was

50 performed with mt = 5000 GeV, and cross sections of the two jet processes converge towards the effective limit as expected. The implementation of scattering amplitudes of the Φjjj production into the framework of VBFNLO was checked against MadGraph [31] within the ef- fective limit by comparing amplitudes at individual phase space points. The −16 comparisons for the Higgs-bosons HSM and A agreed at the 10 level. For large top-quark mass values, mt = 5000 GeV, the full loop contributions were checked only for sub-processes containing quark and gluons as external par- ticles (see Eqs. (3.46) and (3.47)). As expected, the convergence towards the effective limit could also be confirmed numerically for a selection of randomly generated phase space points and fully integrated cross sections.

51

Chapter 5 Applications to LHC physics

5.1 Introduction

The numerical analysis of the Φ+2 and Φ+3 jet cross section was performed with the gluon fusion part GGFLO of the parton level Monte Carlo program VBFNLO [4], using the CTEQ6L1 [40] set for parton-distribution functions. In order to prevent soft or collinear divergencies in the cross sections, a minimal set of acceptance cuts has to be introduced [1]

p > 20 GeV, η < 4.5, R > 0.6 , (5.1) Tj | j| jj where pTj is the transverse momentum of a final state parton and Rjj de- scribes the separation of the two partons in the pseudo-rapidity η versus azimuthal-angle plane

2 2 Rjj = ∆ηjj + φjj , (5.2) q with ∆η = η η and φ = φ φ . These cuts anticipate LHC detec- jj | j1 − j2| jj j1 − j2 tor capabilities and jet finding algorithms and will be called ”inclusive cuts” (IC). For weak-boson fusion (WBF) studies, gluon-fusion induced processes can be suppressed by the use of an additional set of selection cuts (WBFC) [2]

∆η = η η > 4.2 , η η < 0 , m > 600 GeV . (5.3) jj | j1 − j2| j1 · j2 jj The WBFC allow only well separated tagging jets, lying in opposite detector hemispheres and having large dijet invariant mass. Furthermore, a one- loop running of αs was performed with a fixed value at the MZ scale, that is αs(MZ )=0.13. If not specified explicitely, for the top-quark mass the

53 updated value of mt = 171.3 GeV [41] is used in Yukawa couplings as well as in loops with virtual top-quarks. In the case of bottom-loops, a running Yukawa coupling is taken into account with the Higgs-mass as reference scale. Within the Higgs-mass range of 100-600 GeV, the bottom-quark mass is 33-42 % smaller than, the pole mass of 4.79 GeV [41] used in the loop propagators.

The evolution of mb up to a reference scale µ can be expressed as

c αs(µ)/π mb(µ)= mb (mb) . (5.4) c αs(mb)/π with mb (mb)=4.2 GeV, as derived from the relation between pole mass and MS-bar mass. For the coefficient function c the five flavor approximation [16] is given by

12 23 23 c(x)= x 1+1.175x +1.501x2 +0.1725x3 . (5.5) 6     Unless specified otherwise, the factorization scales for two and three jet pro- cesses are set to

2j 3j 1/3 µf = √pT 1 pT 2 and µf =(pT 1 pT 2 pT 3) , (5.6) while the renormalization scales are fixed by setting [1, 42]

4 2 αs(µR)= αs(pT 1)αs(pT 2)αs(mΦ)and (5.7) 5 2 αs(µR)= αs(pT 1)αs(pT 2)αs(pT 3)αs(mΦ) . (5.8)

5.2 Production of the -odd Higgs boson A CP in association with two jets

The production of the -odd Higgs boson A in association with two jets CP at order α4, proceeds in analogy to the -even Higgs boson H of the s CP SM SM [2, 3]. The Higgs boson A is produced via massive quark loops, for which only the third quark generation is taken into account. Here, for the top-quark mass the value of mt = 172.6 GeV is used. Furthermore, massive squark loops can be safely neglected, because these contributions sum up to zero at amplitude level (see (A.25)).

54 1000 100 qq tan β = 1 qg tan β = 7 gg tan β = 30 100 tan β = 1, IC tan β = 50 10 IC 10 [pb] [pb] σ σ 1 1

0.1

0.1 100 200 300 400 500 600 100 200 300 400 500 600

mA [GeV] mA [GeV]

Figure 5.1: A + 2 jet cross sections as a function of the Higgs boson mass. Left panel: individual contributions to the gluon-fusion process (gg, qg and qq amplitudes) for tan β = 1. Right panel: cross sections of the complete A + 2 jet process for different tan β. All processes include interferences of top- and bottom-quark loops. For both panels the inclusive cuts (IC) of Eq. (5.1) are applied.

10 100 qq tan β = 1 qg tan β = 7 gg 10 tan β = 30 tan β = 1, WBFC tan β = 50 WBFC 1 1 [pb] [pb]

σ σ 0.1

0.01 0.1

0.001 100 200 300 400 500 600 100 200 300 400 500 600

mA [GeV] mA [GeV]

Figure 5.2: A + 2 jet cross sections as functions of the Higgs boson mass but with applied WBFC set (5.3).

Expected cross sections at the LHC are shown in Fig. 5.1, as a function of the Higgs boson mass mA within the minimal cuts of Eq. (5.1). The left

55 panel in Fig. 5.1 shows different cross sections of the individual contribu- tions (gg, qg and qq amplitudes) to the gluon fusion process with tan β = 1. Cross sections for processes containing external gluons are much bigger than those induced by quark-quark contributions. The reason is an increase of the gluon-pdf at small x leading to soft events in the initial state. The mass dependence of the cross section of the complete gluon-fusion process for dif- ferent tan β and with top- and bottom-quark interference is given in the right panel of Fig. 5.1. For small tan β, the cross section induced by a top-quark loop dominates compared to the cross section with bottom-quark loop, which is naturally suppressed by the small Yukawa coupling. The minimal cross section is obtained for tan β 7 due to A A (see Eq. (A.7) and (A.8)). ≈ Ct ≈ Cb In case of large tan β, the bottom-quark contributions dominate. However, they show a much more rapid decrease of the cross section with rising mA, since the suppression scale of loops is now set by the heavy Higgs boson mass instead of the quark mass. The striking peak arises due to threshold enhancement at m 2 m , whereas for the bottom-quark loop dominated H ≈ t process no peak appears within this mass range. To avoid singular increase of the cross section, the threshold enhancement was smeared out by integrat- ing over the expected Breit-Wigner peak of the Higgs. For this purpose, the width was determined by adding partial widths for A gg, τ +τ −, b¯b, tt¯ . →{ } In comparison to Fig. 5.1, subprocesses with external gluons, shown in Fig. 5.2, are strongly suppressed by the WBFC set, because events with jets in the central region are cut away due to the requirement of large dijet invariant mass. Hence the over-all cross section decreases as expected.

The left side of Fig. 5.3 shows the tan β dependence of the cross section for different Higgs-masses. Here, the minimum moves with increasing Higgs- mass to higher values of tan β. On the one hand this behavior is correlated with the steeper fall-off of the bottom-loop dominated contributions to the cross section for large tan β with increasing Higgs mass. And on the other hand, it is simply the fact, that couplings to up-type quarks are weakened and to down-type quarks enhanced by tan β respectively. The presence of the

γ5-matrix in the analytic expression for the fermion loops leads, of course, to a new tensor structure (see Appendix B) and to a normalization of the loops, that gives rise to a (3/2)2 =2.25 bigger cross section compared to the Standard Model [1], if tan β = 1, as given in Fig. 5.3. Up to Higgs-masses

56 1000 mA=120 GeV mA,H=120 GeV, IC mA=200 GeV 100 A, tan β=1 mA=300 GeV A, eff. th. 100 mA=400 GeV H IC H, eff. th.

10 [pb] [pb]

σ σ 10

1

0.1 1 5 10 15 20 25 30 35 40 45 50 100 200 300 400 500 600

tan β mA [GeV]

Figure 5.3: Cross section as a function of tan β for different Higgs masses (left panel) and comparison of cross sections of the -odd CP and -even Higgs coupling in both loop-induced and effective theory CP (right panel). Here, the inclusive cuts (IC) of Eq. (5.1) were applied.

of 160 GeV and for small transverse momenta pTj . mt, the effective La- grangian approximation gives correct results and can be used as a numerically fast alternative for phenomenological studies. The effective Lagrangians for the -even and -odd Higgs bosons are described in more detail in Ap- CP CP pendix E. Furthermore, the enhancement at threshold is more smoother in the case of the -even Higgs boson due to the additional contributing par- CP tial widths of decays to W and Z gauge bosons. For the -odd case, those CP decay modes are forbidden at tree-level and, thus, have not been considered. Effects of bottom-loop dominated Ajj production become also noticeable on the transverse momentum distributions of the accompanying jets. They are clearly visible in Figs. 5.4 and 5.5, where the transverse-momentum distri- butions of the softer and the harder of the two jets are shown for pseudo- scalar Higgs masses mA = 120, 200 and 400 GeV and tan β = 1, 7, 30.

For pTj > mb, the large scale of the kinematics invariants leads to an ad- ditional suppression of the bottom induced sub-amplitudes compared to the heavy quark effective theory. Furthermore, with increasing tan β both dis- tributions fall more steeply, which means, that both tagging jets get softer. Above tan β 30 this evolution stagnates. ≈ 57 0.1 tan β = 1 0.1 tan β = 1 0.1 tan β = 1 tan β = 7 tan β = 7 tan β = 7 tan β= 30 tan β= 30 tan β= 30 0.01 m =120 GeV, IC 0.01 m =200 GeV, IC m =400 GeV, IC A A 0.01 A

(jet) [1/GeV] 0.001 (jet) [1/GeV] 0.001 (jet) [1/GeV]

Tmin Tmin Tmin 0.001 /dp /dp /dp σ 1e-04 σ 1e-04 σ d d d σ σ σ 1/ 1/ 1/ 1e-04 1e-05 1e-05

50 100 150 200 50 100 150 200 50 100 150 200

pTmin(jet) [GeV] pTmin(jet) [GeV] pTmin(jet) [GeV]

Figure 5.4: Normalized transverse-momentum distributions of the softer jet in Ajj production at the LHC, for different tan β and Higgs- mass values The inclusive selection cuts of Eq. (5.1) are applied.

tan β = 1 tan β = 1 tan β = 1 0.01 tan β = 7 0.01 tan β = 7 0.01 tan β = 7 tan β = 30 tan β= 30 tan β= 30

mA=120 GeV, IC mA=200 GeV, IC mA=400 GeV, IC 0.001 0.001 0.001 (jet) [1/GeV] (jet) [1/GeV] (jet) [1/GeV] 1e-04 Tmin Tmin 1e-04 Tmin /dp /dp /dp σ σ σ

d d d 1e-04

σ 1e-05 σ σ 1/ 1/ 1/ 1e-05

1e-06 1e-05 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500

pTmax(jet) [GeV] pTmax(jet) [GeV] pTmax(jet) [GeV]

Figure 5.5: Normalized transverse-momentum distributions of the harder jet in Ajj production at the LHC, for different tan β and Higgs- mass values The inclusive selection cuts of Eq. (5.1) are applied.

The azimuthal angle distribution between the more forward and more back- ward of the two tagging jets provides information about the -property of CP the Higgs coupling. Here, for the -odd case the maxima of the distribu- CP tion are located at φ 90 degrees in contrast to the -even coupling. jj ≈ ± CP A direct comparison is shown in the left panel of Fig. 5.7. The calculation

58 0.004 0.004 0.0035 0.0035 0.0035 0.003 0.003 0.003

jj jj jj 0.0025 φ 0.0025 φ φ

/d /d 0.0025 /d σ σ σ

d 0.002 d d 0.002 σ σ 0.002 σ 1/ 1/ 1/ 0.0015 0.0015 0.0015 β β β 0.001 tan = 1 tan = 1 0.001 tan = 1 tan β = 7 0.001 tan β = 7 tan β = 7 tan β = 30 tan β = 30 tan β = 30 0.0005 0.0005 0.0005 mA=120 GeV, ICphi mA=200 GeV, ICphi mA=400 GeV, ICphi

-150 -90 -45 0 45 90 150 -150 -90 -45 0 45 90 150 -150 -90 -45 0 45 90 150 φ φ φ jj jj jj

Figure 5.6: Azimuthal-angle distributions between the two final jets of the -odd Higgs boson for different Higgs-masses and tan β values. CP Here, the ICphi set of Eq. (5.9) is used. was carried out with a modified inclusive cuts set (ICphi) [43]:

p > 30 GeV, η < 4.5, R > 0.6; ∆η =3 . (5.9) Tj | j| jj jj

The additional ∆ηjj cut is necessary to get that distinct shape for the φjj- distribution, which is shown in figures 5.6 and 5.7. For a relatively light pseudo-scalar Higgs boson and large tan β, the softer transverse momentum distribution of the Higgs leads to kinematical distortions of the φjj distri- bution: at φ 0 the Higgs recoils against two jets and hence must have jj ≈ pTH > 60 GeV, and this high pT -scale leads to an additional suppression as compared to the φ 180 degree case where transverse momentum jj ≈ ± balancing of the jets does allow pT,Φ = 0.

5.3 Production of a violating Higgs boson CP Φ in association with two jets

The azimuthal angle distribution for the -even and -odd Higgs bosons CP CP were investigated at first in the context of effective theories [44]. As men- tioned already in the previous chapter, both distributions are clearly distin- guishable by the position of the extreme values. Furthermore, it is interesting to investigate models with -violating Higgs sector and see the impact of CP 59 A / 2.25, tan β = 1 tan β = 1 10 0.005 H tan β = 7 β mA=120 GeV, ICphi tan = 30 8 0.004 mφ=120 GeV, ICphi

jj φ /d [fb]

jj 6 0.003 σ φ d /d σ σ d 1/ 4 0.002

2 0.001

0 0 -150 -90 -45 0 45 90 150 -150 -90 -45 0 45 90 150 φ φ jj jj

Figure 5.7: The azimuthal-angle distribution of the -odd and - CP CP even coupling (left panel). Interference pattern within a toy model with -violating Higgs-sector for different tan β (right panel). The CP -odd coupling was adjusted to the -even one by a factor of 2/3. CP CP Here, the ICphi set (5.9) was used.

-effects on the azimuthal angle distribution. Such effects can appear e.g. CP in the cMSSM, where complex phases and loop effects cause mixing of scalar and pseudo-scalar Higgs bosons to new -eigenstates. But these analyses CP are quite involved and lie beyond the scope of this work. Unfortunately the original azimuthal angle variable, defined by ~p ~p cos φ = Tj1 · Tj2 , (5.10) jj ~p ~p | Tj1 || Tj2 | is not sensitive to these effects and would give a flat distribution correspond- ing to that one of the electro-weak boson fusion [45]. A redefinition of the azimuthal angle variable was introduced in Refs. [43, 46]. To mimic vio- CP lation in the Higgs sector, one can combine simply the -even and -odd CP CP Yukawa couplings together y m = q q iC + C γ with y = cot β,y = tan β (5.11) YΦ v H A 5 u d and investigate the impact on the
azimuthal angle distribution. The scaling factor yq allows to change the strength of the new coupling to up-type and down-type quarks with tan β, similar to the case of the A boson. The parame- ters C and C denote the magnitudes of the -odd and -even coupling A H CP CP 60 respectively. Depending on the value of these parameters, the azimuthal- angle distribution moves between these curves and characterizes the grade of -violation. An example, which also includes interference effects between CP bottom- and top-quark loops, is shown for different tan β in the right panel of Fig. 5.7. The value of CA was fixed at CA =2/3 to yield equal strengths for both Higgs-couplings. However, there are additional distortions of the azimuthal angle distributions, which can again be explained by kinematical effects due to transverse momentum balancing of the two jets and the Higgs boson.

5.4 Production of the -even Higgs bosons CP h and H in association with two jets with additional squark contributions

Numerical investigation of the -even H within the SM was already per- CP SM formed in Ref. [2]. For the contributing subprocesses, only contributions mediated by top-quark loops were taken into account. Loops with bottom- quarks were neglected, because of the missing enhancement of the Yukawa coupling by further model parameters, like tan β, e.g. in a general 2HDM or the MSSM. As already mentioned, the two Higgs-doublet structure of the MSSM provides two -even Higgs bosons with different masses, but CP with the same tensor structure corresponding to that of the SM Higgs boson

HSM. However, their Yukawa couplings to up-type and down-type quarks are modified by additional factors containing the mixing angle α of the neu- tral components of both Higgs-doublets and the angle β coming from the ratio of the two vacuum expectations values v1 and v2. This means, that contributions mediated by bottom-quark loops can gain sizeable enhance- ment. Furthermore, all massive supersymmetric partners of the SM particles can couple to the Higgs bosons and, hence, give further contributions. The new couplings involve additional model parameters, like the trilinear cou- plings At, Ab and supersymmetric Higgs-mass parameter µ, whose values are restricted by given experimental exclusion limits [41]. Here, in this analy- sis only contributions with squarks of the third generation are taken into account. This third generation of squarks is somewhat special, due to the

61 non-negligible masses of their fermion partners. The off-diagonal L-R mixing terms (see chapter 2.3) in the squark mass matrix are large, because of their proportionality to mb or mt. They provide a large splitting, especially in the stop case. The bottom and top squarks manifest themselves physically as the mass eigenstates ˜b1,2 and t˜1,2. This makes t˜1 the lightest squark and hence interesting for the analysis. The calculation of squark mixing angles

θq˜ is explained briefly in chapter 2.3. A further property of supersymmetry is, that contributions with SM-like particles are canceled by contributions containing supersymmetric partners, which carry opposite sign factors. This effect is investigated below in more detail for the light Higgs boson h and heavy Higgs boson H. The evaluation was done using the following values as input parameters:

t b A [GeV ] A [GeV ] µ [GeV ] mA [GeV ] . 1000 1500 150 200

Based on the statement, that the top squark is possibly the lightest squark, the t˜1-mass was chosen as the scan parameter for the analysis. All other masses of the remaining top and bottom squarks were calculated with the FeynArts, FormCalc [37, 38, 39] package using the input parameters men- tioned above. Depending on the values for the squark masses and the param- eter tan β, loop corrected masses for the Higgs bosons h and H were calcu- lated with the help of FeynHiggs [47, 48, 49, 50]. The masses of both Higgs bosons alter very slightly, although the contributions mediated by squark- loops change continuously during the scanning procedure. They are shown in Fig. 5.8. Hence, only marginal changes of the cross section are expected over the whole scanning range. The mass m 102 GeV is excluded by h ≈ experiment and is used only for the purpose of illustration. The final evalua- tion was done with the gluon fusion part GGFLO of the VBFNLO program. Expected cross sections for the production of the light and the heavy Higgs bosons are shown in Fig. 5.9 as a function of the t˜1-mass for different values of tan β within the minimal cuts of Eq. (5.1). For the light Higgs boson h, one can see very clearly the expected impact of the contributions mediated by squark-loops (left panel of Fig. 5.9). In the range between 180-400 GeV the cross section is strongly suppressed as a

62 140

135 h 220 H tan β = 3 tan β = 3 130 tan β = 7 tan β = 7 tan β = 50 tan β = 50 125 215

120 [GeV] [GeV]

h H 210 115 m m

110 205 105

100 200 95 180 400 800 1200 1600 400 800 1200 1600

mstop 1 [GeV] mstop 1 [GeV]

Figure 5.8: Loop-corrected Higgs masses of h and H as a function of

the t˜1-mass for different values of tan β.

3.5 18 H, IC 3 tan β = 3 16 tan β = 7 2.5 tan β = 50 14 h, IC 2 tan β = 3 12 [pb] β [pb] σ tan = 7 σ 1.5 tan β = 50 10 1

8 0.5

6 0 180 400 800 1200 1600 180 400 800 1200 1600

mstop 1 [GeV] mstop 1 [GeV]

Figure 5.9: Full cross section with t and b quark as well as t˜1,2 and ˜b1,2 squark contributions as a function of t˜1-mass for different values of tan β and Higgs-masses shown in Fig. 5.8. For both panels the inclusive cuts (IC) of Eq. (5.1) were applied.

result of strong cancellations between SM-like particle and SUSY-like parti- cles. For masses beyond 600 GeV, the scalar loops decouple or rather become negligible and the evolution of the cross section stagnates with contributions

63 mediated only by top- and bottom-quarks. To get a feeling for the influence of tan β on the Yukawa and scalar couplings, the cross sections of individual loop-contributions and the full result including interference effects of all am- plitudes are shown in the upper table 5.1 for a t˜1-mass of 180 GeV, which corresponds to the first parameter set of Fig. 5.9:

σ(h) [fb]

˜ ˜ tan β t b t˜1 t˜2 b1 b2 +int.

3 16058,2 20,8 2717,5 89,7 0.5 2, 9 10−2 P7263.2 · 7 15135,1 17,1 2964,6 84,4 0.5 1, 0 10−3 6242.1 · 50 15092,2 16,4 3236,9 84,0 1,9 1,2 5746.4

σ(H) [fb]

3 1585,8 5,4 919,3 13,8 0,3 0,9 213.2

7 385,7 36,5 413,0 4,8 1,9 2,5 46.7

50 8,3 1980,7 140,5 1.0 117,7 229,3 1963.5

Table 5.1: Cross sections of the light Higgs h (top) and heavy Higgs H (bottom) ordered by individual contributions with different loop-masses and values of tan β. Total cross sections contain interference effects of all amplitudes. The inclusive selection cuts of Eq. (5.1) were applied.

With increasing tan β, only contributions induced by ˜b2-loop gain an appre- ciable enhancement. Furthermore, loops with t˜1 provide the biggest con- tribution in the squark-sector up to high values of tan β, because of their relative small mass. However, the influence of tan β is quite slight. Also the contribution of t˜2 remains almost constant and yields a small cross section, which is an effect of the bigger loop-mass. Amplitudes with top- and bot- tom quarks hardly get modifications, because the ratio of the trigonometric functions depending on α and β inside the Yukawa couplings (see App. A.1)

64 stay, roughly speaking, in equilibrium, a sign of h decoupling at large mA. Similar evolution of the cross sections for the heavy Higgs boson H is found in the right panel of Fig. 5.9, except for high values of tan β. Generally, the cross section for the Hjj production turns out to be smaller in comparison to the light Higgs boson h. Namely it gets additional suppression due to the bigger mass. The cross sections show an equal behavior for masses beyond 600 GeV, where contributions with squarks get strongly suppressed indepen- dently of tan β by rising squark-masses in the loop-propagators. To get a better understanding of the behavior of the cross section for tan β = 50, val- ues of σ for the first parameter set corresponding to the t˜1-mass of 180 GeV and for different tan β are shown in the lower table 5.1. Here, contributions with top-quarks are strongly suppressed in contrast to contribution mediated by bottom-quark loops for raising values of tan β. This behavior is similar to that of the -odd Higgs boson A. The same happens for up-type and CP down-type squarks. For high values of tan β, amplitudes with bottom-loops dominate, however, contributions induced by all squark-loops are still too small to compensate the contribution given by the quark-loops.

The result is an almost flat distribution over the whole t˜1-mass range. By the way, full cross sections show also a distinct minimum for tan β = 7.

0.1 h, IC h, IC β β 0.01 tan = 3 tan = 3 tan β = 7 tan β = 7 tan β = 50 0.01 tan β = 50 (jet) [1/GeV] (jet) [1/GeV]

Tmax 0.001 Tmin /dp /dp σ σ 0.001 d d σ σ 1/ 1/ 1e-04

1e-04

1e-05 100 200 300 400 500 100 200

pTmax(jet) [GeV] pTmin(jet) [GeV]

Figure 5.10: Normalized transverse-momentum distributions of the harder and softer jet in hjj production at the LHC, for different tan β and Higgs-masses shown in Fig. 5.8.

65 0.1 H, IC H, IC 0.01 tan β = 3 tan β = 3 β β tan = 7 0.01 tan = 7 tan β = 50 tan β = 50 (jet) [1/GeV] (jet) [1/GeV] 0.001 Tmax Tmin 0.001 /dp /dp σ σ d d σ σ

1/ 1e-04 1/ 1e-04

1e-05 1e-05 100 200 300 400 500 100 200

pTmax(jet) [GeV] pTmin(jet) [GeV]

Figure 5.11: Normalized transverse-momentum distributions of the harder and softer jet in Hjj production at the LHC, for different tan β and Higgs-masses shown in Fig. 5.8.

The reasons for this minimum are already small contributions provided by quark-loops and further suppression by amplitudes with squark-loops. Complementary to the table 5.1, distributions for the transverse momenta of the accompanying jets and azimuthal angle are shown in Figs. 5.10, 5.11 and Fig. 5.12 respectively. The almost equal shapes for the production of the light Higgs boson h with fermionic and sfermionic contributions for different tan β confirm the fact, that there is no significant influence of the parameter tan β. On the contrary, the heavy Higgs boson H is much more sensitive to tan β and shows a similar behavior as the -odd Higgs boson A. With increasing CP values of tan β, the transverse momentum distribution of the softer and the harder of the two jets fall more steeply. Furthermore, the softer transverse momentum distributions of H leads also to kinematical distortion of the azimuthal angle distribution, which is here very clearly visible at φ 90. jj ≈ ± Ones has to keep in mind, that contributions of sfermionic loops provide an additional suppression of the cross section, which becomes noticeable on the behavior of the φjj-distribution for tan β = 7 in comparison to the Higgs boson A.

66 0.004 0.0045

0.004 0.0035

0.0035 0.003

jj 0.003 jj 0.0025 φ φ /d /d

σ 0.0025 σ

d d 0.002 σ σ

1/ 0.002 1/ 0.0015 0.0015 h, ICphi H, ICphi tan β = 3 0.001 tan β = 3 0.001 tan β = 7 tan β = 7 tan β = 50 tan β = 50 0.0005 0.0005

0 0 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 φ φ jj jj

Figure 5.12: Azimuthal-angle distributions between the two final jets of the -even Higgs bosons h and H for different tan β and Higgs- CP masses shown in Fig. 5.8. Here the ICphi set is used.

5.5 Production of the Higgs bosons Φ in as- sociation with three jets

As already discussed in the previous chapters, the azimuthal angle correla- tion φjj between the accompanying jets is a sensitive measure of the tensor structure for the Higgs couplings to electro-weak bosons or gluons. Espe- cially in the case of Higgs boson production via gluon fusion, characteris- tic shapes of the φ -distribution for -even and -odd Higgs couplings, jj CP CP shown in Fig. 5.7, allow to discriminate between the SM or extensions e.g. the MSSM. A further aspect of interest is the modification of the azimuthal angle correlation by the emission of additional gluons. Former investigations with showering and hadronisation provided a strong de-correlation between the tagging jets in Higgs +2 jet production [51]. The de-correlation effects, however, were disproportionately illustrated due to approximations in the parton-shower. More recent analyses [52, 53] show, that after separation of hard radiation and showering effects with subsequent hadronisation, the

φjj-correlation survives with minimal modifications. Similar results were ob- tained by a parton level calculation with NLO corrections [21] to the Hjj process in the framework of the effective Lagrangian.

67 0.0055 0.007 0.005 qq->qqA 0.006 qq->qqH 0.0045 qq->qqgA qq->qqgH 0.004 ICphi 0.005 ICphi

jj 0.0035 jj φ φ

/d /d 0.004 0.003 σ σ d d

σ 0.0025 σ 0.003 1/ 1/ 0.002 0.0015 0.002 0.001 0.001 0.0005 0 0 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 φ φ jj jj

Figure 5.13: Normalized azimuthal-angle distributions for the pro- duction of the -odd and -even Higgs bosons in the sub-processes CP CP qq qqΦ, qq qqgΦ and crossing related sub-processes. The ICphi → → set of Eq. (5.9) is used.

The calculation and implementation of the process Φjjj with full massive loops, introduced in chapter 3.4, allows also among other things, the inves- tigation of de-correlation effects beyond present frameworks, which only use the effective Lagrangian approach. As a primal result, the azimuthal-angle distributions of the processes qq → qqΦ and qq qqgΦwith Φ = H , A are shown in Fig. 5.13 for the modified → SM inclusive cuts of Eq. (5.9). Furthermore, crossing related sub-processes were also taken into account (see also Eq. (3.46) ). All processes contain top-quark triangle loops and the three jet process even additional Feynman diagrams with box-loop insertions. In order to simplify matters, the value of tan β =1 is taken in the coupling of the -odd Higgs boson A. CP Both φjj distributions confirm de-correlation effects, when the process Φjj is extended by a further jet. One can see, that the dips at φ 0 for jj ≈ the -odd A and φ π/2 for the -even H are shallower then in CP jj ≈ ± CP SM the corresponding process with two jets. But these first results have to be handled with care, because they show contributions, which only represent a small fraction of the total process. The additional sub-processes qg qggΦ, → gg gggΦ and crossing related sub-processes, that are expected to give the → 68 main contributions, however, could not be taken into account. The ampli- tudes of these contributions contain pentagons and even hexagons and make the numerical analysis quite involved. Due to the extended phase-space in the three jet process, and hence, increased amount of possible invariants en- tering the tensor reduction, numerical instabilities appeared at the level of five-point functions from rank two up to four. They deformed different distri- butions, which are the subjects of this investigation, and made the analysis impossible. A further analysis of the involved phase-space regions is neces- sary to have a better control over the numerical evolution of the five-point functions, that are also important as input for the six-point functions. Also further comparisons with the two-jet processes have to be performed, because the numerical evaluation of these processes caused no problems, although all versions of five-point functions up to rank four were applied. The produced results were already shown in the analysis chapters for the Higgs production processes with two jets, using at least 224 randomly generated phase-space points.

69

Chapter 6 Conclusions

Searching for the Higgs boson within the framework of the Standard Model and its supersymmetric extensions is one of the main tasks of the LHC. The two main sources of Higgs plus two jet events are weak-boson fusion and gluon fusion. Hence, a precise description of both processes is needed, in or- der to separate them from each other. This thesis present the calculation of scattering amplitudes for the production of neutral Higgs bosons within the MSSM via gluon fusion with a two and three jet finale-state. The scattering amplitudes for this processes are induced by triangle-, box-, pentagon- and hexagon-loop diagrams at leading order. In this connection analytical ex- pressions of loop-topologies were evaluated for scalar and fermionic particles as well as for -odd and -even Higgs couplings. The tensor reduction CP CP for triangle- and box-loops was performed via Passarino-Veltman reduction, while for the pentagons and hexagons the Denner-Dittmaier algorithm was applied to obtain numerically stable results. In the first part of the Chap- ter 3, a detailed description of the analytical expressions of φjj scattering amplitudes for sub-processes qq qqφ, qg qgφ, gg ggφ and crossing → → → related sub-processes was presented. The second part of Chapter 3 described in a detailed way the calculation of scattering amplitudes for a Higgs plus three-jet final state. Chapter 4 was devoted to a brief description of analyti- cal and numerical consistency checks of the calculation and implementation into the Monte Carlo programVBFNLO. In Chapter 5 fully integrated cross sections of the Ajj production with interference effects between top- and bottom-loops were compared for different values of tan β and masses mA of the -odd Higgs boson. Furthermore, it was shown, that small quark CP masses in the bottom-loops provide a softening effect on the transverse mo- mentum distribution of the accompanying jets, which cannot be described

71 within the framework of the effective Lagrangian. Distortion effects were also observed in the azimuthal angle correlation of both jets in the bottom-loop dominated Ajj production. Moreover, a pronounced difference between Ajj and Hjj production in the azimuthal angle distribution between the two jets could also be observed. This effect provides the possibility to determine the -properties of the produced Higgs boson at the LHC. In addition to that, CP effects of -violating couplings were simulated, which allow to investigate CP scenarios with a -violating Higgs sector. Furthermore, interference effects CP of contributions containing squark- and quark-loops in the production of the -even Higgs bosons h and H plus two jets were investigated. Performed CP scans for a set of MSSM-parameters confirmed the expected cancellations be- tween the sfermionic and fermionic contributions, which resulted in a strong reduction of the integrated cross sections. Based thereupon, effects of dif- ferent values of tan β, especially for the heavy Higgs boson H, on azimuthal angle and transverse momentum distributions could be illustrated. Finally, preliminary results of the φjjj implementation were presented for a small set of sub-processes. Here, de-correlation effects of the two-jet azimuthal angle distribution were presented, which were an effect of the third jet produced by the additional gluon in the final state.

72 Appendix A Higgs vertices to fermions and sfermions

A.1 Higgs couplings to fermions

The Lagrangian for Higgs-fermion Yukawa interactions has the following form

= φF F φ , (A.1) LY Cf f f φ=Xh,H,A fX=u,d in which the corresponding factors φ are given by Cf

HSM , CP-even :  m  SM = i f (A.2) Cu,d v h, CP-even, u-type :  m cos α  h = i u (A.3) Cu v sin β

h, CP-even, d-type :

 m sin α h = i d (A.4) Cd − v cos β

H, CP-even, u-type :

 m sin α  H = i u (A.5) Cu v sin β

73 H, CP-even, d-type :  m cos α H = i d (A.6) Cd − v cos β

A, CP-odd, u-type :  m  A = u cot βγ (A.7) Cu − v 5 A, CP-odd, d-type :  m  A = d tan βγ (A.8) Cd − v 5

74 A.2 Higgs couplings to sfermions

General expressions for the couplings of neutral Higgs bosons to squarks can be looked up in Ref. [10]. Assuming no flavor mixing, these expressions were evaluated using parameters of the MSSM. Furthermore, abbreviations are introduced for expressions containing the mixing angle α between the neutral components of both Higgs doublets, β coming from the ratio of the two vacuum expectations values, the electro-weak mixing angle θW and squark mixing angles θf˜: s = sin α, β , c = cos α, β , t = tan θ , {α,β} { } {α,β} { } W W sα+β = sin(α + β) , cα+β = cos(α + β) ,

sθf˜ = sin θf˜ , cθf˜ = cos θf˜ . The Lagrangian for Higgs-sfermion interactions can be written as

φ ˜∗ ˜ φ f˜ = ˜ φfi fi , L Cfi φ=h,H,A, ˜ ˜ X f=˜Xui,dj φ in which the corresponding factors ˜ are given by Cfi h, CP-even, u-type, 1 :   h 2 u 2 u˜1 = cθu˜ sθu˜ muA mu cα muµcθu˜ sθu˜ sα C vsβ " − − #   M 2 1 4 + W c2 1 t2 + s2 t2 s (A.9) v θu˜ − 3 W 3 θu˜ W α+β h i h, CP-even, u-type , 2 :
  h 2 u 2 u˜2 = cθu˜ sθu˜ muA + mu cα muµcθu˜ sθu˜ sα C −vsβ " − #   M 2 1 4 + W s2 1 t2 + c2 t2 s (A.10) v θu˜ − 3 W 3 θu˜ W α+β h i h, CP-even, u-type , L :

2  h 2 2 M W 1 2 u˜L mucα + 1 tW sα+β (A.11) C − vsβ v − 3
75 h, CP-even, u-type, R :

2  h 2 2 MW 4 2 u˜R mucα + tW sα+β (A.12) C − vsβ v 3 H, CP-even, u-type, 1 :   H 2 u 2 u˜1 = cθu˜ sθu˜ muA mu sα + muµcθu˜ sθu˜ cα C vsβ " − #   M 2 1 4 W c2 1 t2 + s2 t2 c (A.13) − v θu˜ − 3 W 3 θu˜ W α+β h i H, CP-even, u-type , 2 :
  H 2 u 2 u˜2 = cθu˜ sθu˜ muA + mu sα + muµcθu˜ sθu˜ cα C −vsβ " #   M 2 1 4 W s2 1 t2 + c2 t2 c (A.14) − v θu˜ − 3 W 3 θu˜ W α+β h i H, CP-even, u-type , L :

2  H 2 2 MW 1 2 u˜L = musα 1 tW cα+β (A.15) C −vsβ − v − 3
H, CP-even, u-type, R :

2  H 2 2 MW 4 2 u˜R = musα tW cα+β (A.16) C −vsβ − v 3 h, CP-even, d-type, 1 :

 2  h = c s m Ad m2 s m µc s c d˜1 θd˜ θd˜ d d α d θd˜ θd˜ α C −vcβ " − − #   2 MW 2 1 2 2 2 2 cθ ˜ 1+ tW + sθ ˜tW sα+β (A.17) − v d 3 3 d h i h, CP-even, d-type , 2 :

 2  h = c s m Ad + m2 s m µc s c d˜2 θd˜ θd˜ d d α d θd˜ θd˜ α C vcβ " − #   2 MW 2 1 2 2 2 2 sθ ˜ 1+ tW + cθ ˜tW sα+β (A.18) − v d 3 3 d h
i 76 h, CP-even, d-type, L :

2  h 2 2 M W 1 2 = m sα 1+ t sα+β (A.19) d˜L d W C vcβ − v 3
h, CP-even, d-type, R :

2  h 2 2 M W 2 2 d˜ = mdsα tW sα+β (A.20) C R vcβ − v 3 H, CP-even, d-type, 1 :

 2  H = c s m Ad m2 c + m µc s s d˜1 θd˜ θd˜ d d α d θd˜ θd˜ α C vcβ " − #   2 MW 2 1 2 2 2 2 + cθ ˜ 1+ tW + sθ ˜tW cα+β (A.21) v d 3 3 d h i H, CP-even, d-type , 2 :

 2  H = c s m Ad + m2 c + m µc s s d˜2 θd˜ θd˜ d d α d θd˜ θd˜ α C −vcβ " #   2 MW 2 1 2 2 2 2 + sθ ˜ 1+ tW + cθ ˜tW cα+β (A.22) v d 3 3 d h i H, CP-even, d-type , L :

2  H 2 2 MW 1 2 = m cα + 1+ t cα+β (A.23) d˜L d W C −vcβ v 3
H, CP-even, d-type, R :

2  H 2 2 M W 2 2 = m cα + t cα+β (A.24) d˜R d W C −vcβ v 3 (A.25) A, CP-odd, squarks : i  All calculated topologies restrict the coupling squarks to reside either in one of the L,R or 1,2 bases. The property of this coupling implies, that all contributions to the cross section for the -odd Higgs boson CP A, containing loop topologies with squarks, are zero at amplitude level. For more details see the Appendix A.8 p. 399 of Ref. [8].

77

Appendix B Loop tensors

B.1 Three-point functions (Triangles)

B.1.1 Fermion-triangle with Higgs vertex

q1 k q1 k + q1 + q2 φ φ µ1µ2 µ1µ2 k + q1 Tφ,1,f k + q2 Tφ,2,f

q2 k + q1 + q2 q2 k

Figure B.1: Three-point functions connected by charge-conjugation.

The generic three-point functions for fermionic triangle graphs with Higgs vertex and opposite loop momentum, depicted in Fig. B.1, have the following expressions,

d4k k + m k + q + m T µ1µ2 (q , q , m )= tr f γµ1 1 f γµ2 Φ,1,f 1 2 f iπ2 k2 m2 (k + q )2 m2 Z " − f 1 − f k + q + m 12 f , (B.1) × (k + q )2 m2 VΦ 12 − f # d4k k + m k + q + m T µ1µ2 (q , q , m )= tr f γµ2 2 f γµ1 Φ,2,f 1 2 f iπ2 k2 m2 (k + q )2 m2 Z " − f 2 − f 79 k + q + m 12 f , (B.2) × (k + q )2 m2 VΦ 12 − f # where q1, q2 are outgoing gluon momenta and q12 = q1 + q2. The factor denotes the property of the Higgs coupling: = γ5 for -odd and VΦ CP VA CP = 1 for -even Higgs boson. Using the charge conjugation matrix VHSM ,h,H CP Cˆ,

Cγˆ Cˆ−1 = γT , Cγˆ Cˆ−1 = γT with Cˆ = γ0γ2 , Cˆ2 = 1 (B.3) µ − µ 5 5 and the property of the trace tr (γµ1 )T (γµ2 )T . . . = tr ...γµ2 γµ1 , one can derive (Furry’s theorem [29])    

T µ1µ2 (q , q , m )= T µ1µ2 (q , q , m ) T µ1µ2 (q , q , m ) . (B.4) Φ,1,f 1 2 f Φ,2,f 1 2 f ≡ Φ,f 1 2 f

Hence, the relative sign of charge-conjugated graphs changes with the number of fermion-gluon vertices. From this it follows for the color structure

a1 a2 µ1µ2 a2 a1 µ1µ2 tr t t TΦ,1,f (q1, q2, mq)+tr t t TΦ,2,f (q1, q2, mf )

a1a2 µ1µ2  =δ TΦ,f (q1, q2, mf ) ,  (B.5)

a1 a2 1 a1a2 in which the identity tr t t = 2 δ was used.  

B.1.2 Fermion-triangle with -odd Higgs vertex CP The evaluation of the Dirac trace with = γ5 yields VA

µ1µ2 2 µ1µ2q1q2 TA,f (q1, q2, mf )=4 mf ε C0(q1, q2, mf ) . (B.6)

µ1µ2q1q2 Here, C0 denotes the scalar three-point function and ε is the total anti-symmetric tensor (Levi-Civita symbol) contracted with attached gluon µ1µ2 momenta q1 and q2. The tensor TA,f (q1, q2, mf ) is UV-finite. Hence, no difficulties arise with definition of the γ5 Dirac-matrix in d dimensions.

80 B.1.3 Fermion-triangle with -even Higgs vertex CP The Relations of Eqs. (B.3), (B.4) and (B.5) hold also here, but with γ5 replaced by 14. The calculation of the Dirac trace leads to [2]

µ1µ2 2 2 2 2 µ1µ2 T (q1, q2, mf )=4 m FT q , q , (q1 + q2) q1 q2 g HSM ,h,H,f f 1 2 · 1 2 h 1 2 qµ qµ + F q2, q2, (q + q )2 q2 q2
gµ1µ2 q2 qµ qµ − 1 2 L 1 2 1 2 1 2 − 1 2 2 q2 qµ1 q
µ2 + q q qµ1 qµ2 ,
(B.7) − 2 1 1 1 · 2 1 2
i with the form factors

1 F q2, q2, (q + q )2, m = L 1 2 1 2 f −2 det Q2 (
2 3q1q2 (q1 + q2) 2 · B0(q1, mf ) B0(q1 + q2, mf ) × " − det 2 # − Q   2 3q1q2 (q1 + q2) + 2 · B0(q2, mf ) B0(q1 + q2, mf ) " − det 2 # − Q   3q2q2 (q + q )2 4m + q2 + q2 +(q + q )2 1 2 · 1 2 − f 1 2 1 2 − det " Q2 #

C0(q1, q2, mf ) 2 , (B.8) × − )

2 2 2 1 2 FT q1, q2, (q1 + q2) , mf = (q1 + q2) B0(q1, mf ) −2 det 2 Q n + B (q , m ) 2 B (q
+ q , m ) 2 q q C (q , q , m ) 0 2 f − 0 1 2 f − 1 · 2 0 1 2 f + q2 q2 B (q , m ) B (q , m ) q q F .  (B.9) 1 − 2 0 1 f − 0 2 f − 1 · 2 L

o 2 Here, det = q2 q2 q q stands for the Gram determinant. The Q2 1 2 − 1 · 2 shortcuts B and C denote scalar two- and three-point functions. Emerging 0 0
−1 ǫ -poles from the B0-functions cancel each other at intermediate steps, so µ1µ2 that the tensor THSM ,h,H,f (q1, q2, mf ) remains UV-finite. For more details see Ref. [2].

81 q1 k q1 k + q1 + q2 φ φ T µ1µ2 T µ1µ2 k + q1 φ,1,f˜ k + q2 φ,2,f˜

q2 k + q1 + q2 q2 k

Figure B.2: Triangle-loops with opposite fermion flow

B.1.4 Sfermion-triangle with -even Higgs vertex CP The generic three-point functions for sfermionic triangle graphs with - CP even Higgs vertex and opposite loop momentum, shown in Fig. B.2, have the following expressions

µ1µ2 T Φ,1,f˜(q1, q2, mf˜) 4 µ1 µ2 d k 2 k + q1 2(k + q1)+ q2 = 2 , (B.10) iπ 2 2 2 2 2 2 k m (k + q1) m (k + q12 ) m Z − f˜ − f˜ − f˜ h ih ih i

µ1µ2 T Φ,2,f˜(q1, q2, mf˜) 4 µ2 µ1 d k 2 k + q2 2(k + q2)+ q1 = 2 . (B.11) iπ 2 2 2 2 2 2 k m (k + q2) m (k + q12 ) m Z − f˜ − f˜ − f˜ h ih ih i Theq ˜qA˜ -vertex (Eq. (A.24)) gives only a contribution, if both squarks do not reside in the same basis (L/R or 1/2). Hence, for the case depicted in Fig. B.2, all contribution sum to zero at amplitude level. Both tensors can be related via a shift of the loop momentum k k q q , → − − 1− 2 so that the relation of Eq. (B.4) also holds for scalar loops

µ1µ2 µ1µ2 µ1µ2 T ˜(q , q , m ˜)= T ˜(q , q , m ˜) T ˜ (q , q , m ˜) . (B.12) Φ,1,f 1 2 f Φ,2,f 1 2 f ≡ Φ,f 1 2 f

82 The evaluation of the numerator leads to

µ1µ2 T Φ,f˜ (q1, q2, mf˜)

4 µ1 µ2 µ1 µ2 µ1 µ2 µ2 µ1 µ2 µ2 d k 4 k k +2 q1 k +2 k (2 q1 + q2 )+ q1 (2 q1 + q2 ) = 2 iπ k2 m2 (k + q )2 m2 (k + q )2 m2 Z − f˜ 1 − f˜ 12 − f˜ µ1µ2 h ihµ1 µ2 ih µ1 i µ2 =4 C q1, q2, mf˜ +2 q1 C q1, q2, mf˜ +2 C q1, q2, mf˜ 2 q1

µ2 µ1 µ2 µ2 + q2 + q1 (2 q1
+ q2 ) C0 q1, q2, mf˜

µ1
µ2 µ2
µ1 µ2 =2 q2 q2 C12 +2 C22 +2 q1 C12 + C23 + q1 2 q1 C0 +3 C11  h

i +2 C + qµ2 C +2 C + C +2 C 4 gµ1µ2 C . (B.13) 21 2 0 11 12 23 − 24 
h
i The coefficients Cij originate from the Passarino-Veltman tensor reduction procedure [32, 33]. However, the Cµ1µ2 function contains an ǫ−1-pole in di- mensional regularization and hence, it is UV-divergent. This ǫ−1-pole cancel µ1µ2 by addition of a two-point function S (q , q , m ˜) with aq ˜qgg˜ -vertex (see h,H,f˜ 1 2 f Fig. B.3). This graph has a simple analytic expression

k + q + q q1 1 2

φ µ1µ2 Sφ,f˜

q2 k

Figure B.3: Scalar two-point function with a q˜qgg˜ -vertex.

4 µ1µ2 d k gµ1µ2 S (q , q , m ˜)= h,H,f˜ 1 2 f 2 iπ k2 m2 (k + q + q )2 m2 Z − f˜ 1 2 − f˜ µ1µ2 h ih i = g B0 q12, mf˜ (B.14)

−1
The ǫ -pole coming from B0 cancels the corresponding poles originating from both sfermion triangles. Furthermore, the proportionality of theq ˜qgg˜ - vertex to the color factor ta1 , ta2 matches perfectly with Eq. (B.5), so that  83 all three graphs can be summed up to a divergence free contribution

µ1µ2 µ1µ2 µ1µ2 T (q , q , m ˜)= T (q , q , m ˜)+ S (q , q , m ˜) . (B.15) h,H,f˜ 1 2 f h,H,f˜ 1 2 f h,H,f˜ 1 2 f

B.1.5 Sfermion-triangle with -even Higgs vertex and CP q˜qgg˜ -vertex

q1 k q1 k + q12 + q3

µ1µ2µ3 φ µ1µ2µ3 φ q2 TB,φ,1,f˜ q2 TB,φ,2,f˜ k + q12 k + q3 q3 k + q12 + q3 q3 k

Figure B.4: Triangle-loops with opposite fermion flow and q˜qgg˜ - vertex

Theq ˜qgg˜ -vertex (D.5) allows also for triangle-graphs with three attached gluons. They are depicted in Fig. B.4. The integrals can be written in this way

µ1µ2µ3 T (q , q , m ˜) B,Φ,1,f˜ 12 3 f

4 µ1µ2 µ3 d k g 2(k + q12)+ q3 = 2 iπ 2 2 2 2 2 2 k m (k + q12) m (k + q123) m Z − f˜ − f˜ − f˜ µ1µ2 h µ3 ih µ1µ2 µih3 µ3 i =2 g C (q12, q3, mf˜)+ g (2 q12 + q3 )C0(q12, q3, mf˜) , (B.16)

µ1µ2µ3 T (q , q , m ˜) B,Φ,2,f˜ 12 3 f

4 µ1µ2 µ3 d k g 2 k + q3 = 2 iπ 2 2 2 2 2 2 k m (k + q3) m (k + q123) m Z − f˜ − f˜ − f˜ µ1µ2 h µ3 ih µ1µ2 µ3 ih i =2 g C (q12, q3, mf˜)+ g q3 C0(q12, q3, mf˜) . (B.17)

Again, with a simple shift of the loop momentum k k q q one gets → − − 12 − 3 T µ1µ2µ3 = T µ1µ2µ3 T µ1µ2µ3 . (B.18) B,Φ,1,f˜ − B,Φ,2,f˜ ≡ B,Φ,f˜

84 Although it is a triangle-loop, it has a box-like color structure (see Eq. (B.26))

a1 a2 a3 a3 a1 a2 µ1µ2µ3 a1 a2 a3 tr t , t t tr t t , t T ˜ = tr t t t ( − ) B,Φ,f h i h  i n   + tr ta2 ta1 ta3 tr ta3 ta1 ta2 tr ta3 ta2 ta1 T µ1µ2µ3 =0 , (B.19) − − B,Φ,f˜      o in which the cyclic invariance of the trace was used. Hence, these graphs give no contribution.

B.1.6 Sfermion-triangle with -even Higgs vertex and CP two q˜qgg˜ -vertices

q1 k q1 k + q1234 q2 φ q2 φ T µ1µ2µ3µ4 µ1µ2µ3µ4 k + q12 P,φ,1,f˜ k + q34 TP,φ,2,f˜ q3 q3

q4 k + q1234 q4 k

Figure B.5: Sfermionic triangle-loops with opposite fermion flow and pentagon-like color structure

The twoq ˜qgg˜ -vertices allow also for triangle-graphs with four attached gluons (see Fig. B.5)

µ1µ2µ3µ4 T (q , q , m ˜) P,Φ,f˜ 12 34 f d4k gµ1µ2 gµ3µ4 = 2 (B.20) iπ k2 m2 (k + q )2 m2 (k + q )2 m2 Z − f˜ 12 − f˜ 1234 − f˜ h ih ih i This topology is invariant under reversion of the momentum flow. More- over, it has a pentagon-like color structure. For the sum of two graphs with

85 opposite loop momentum, the color factor reads as follows

tr ta1 , ta2 ta3 , ta4 + tr ta3 , ta4 ta1 , ta2 ! h  i h  i µ1µ2µ3µ4 a1 a2 a3 a4 a1 a2 a4 a3 T (q , q , m ˜)=2 tr t t t t + tr t t t t × P,Φ,f˜ 12 34 f  a2 a1 a3 a4 a2 a1 a4 a3 µ1µ2µ3µ4   + tr t t t t + tr t t t t T (q , q , m ˜) , (B.21) P,Φ,f˜ 12 34 f     with

T µ1µ2µ3µ4 (q , q , m )= gµ1µ2 gµ3µ4 C (q , q , m ) . (B.22) P,Φ,f˜ 12 34 f˜ 0 12 34 f˜

B.2 Four-point functions (Boxes)

B.2.1 Fermion-box with Higgs vertex

q q q q 2 k + q1 1 2 k + q23 1

µ1µ2µ3 µ1µ2µ3 k q k + q12 Bφ,1 k k + q3 Bφ,2 + 123

φ φ

q3 k + q123 q3 k

Figure B.6: Four-point functions connected by charge-conjugation.

Four-point functions for the production of the Higgs boson Φ connected via charge-conjugation have the following analytical expressions

4 µ1µ2µ3 d k k + m k +q + m B (q , q , q , m )= tr f γµ1 1 f Φ,1,f 1 2 3 f iπ2 k2 m2 (k + q )2 m2 Z " − f 1 − f k +q + m k +q + m γµ2 12 f γµ3 123 f , (B.23) × (k + q )2 m2 (k + q )2 m2 VΦ 12 − f 123 − f #

86 4 µ1µ2µ3 d k k + m k + q + m B (q , q , q , m )= tr f γµ3 3 f Φ,2,f 1 2 3 f iπ2 k2 m2 (k + q )2 m2 Z " − f 3 − f k +q + m k +q + m γµ2 23 f γµ1 123 f , (B.24) × (k + q )2 m2 (k + q )2 m2 VΦ 23 − f 123 − f # where q1, q2 and q3 are outgoing momenta, qij = qi +qj and qijk = qi +qj +qk. From charge conjugation one gets

µ1µ2µ3 µ1µ2µ3 B (q , q , q , m )= B (q , q , q , m ) Φ,1,f 1 2 3 f − Φ,2,f 1 2 3 f µ1µ2µ3 B (q , q , q , m ) . (B.25) ≡ Φ,f 1 2 3 f

Further two permutations can be achieved by cyclic permutation of (1, 2, 3). The color structure for the sum of the two diagrams is

a1 a2 a3 µ1µ2µ3 a3 a2 a1 µ1µ2µ3 tr t t t BΦ,1,f (q1, q2, q3, mf )+tr t t t BΦ,2,f (q1, q2, q3, mf )

µ1µ2µ3 = tr t
a1 ta2 ta3 tr ta3 ta2 ta1 B (q , q
, q , m ) − Φ,f 1 2 3 f h i i µ
1µ2µ3
= f a1a2a3 B (q , q , q , m ) , (B.26) 2 Φ,f 1 2 3 f in which the following identity was used

1 tr ta1 ta2 ta3 = da1a2a3 + i f a1a2a3 . (B.27) 4

Altogether the sum over the six permutations of boxes is then proportional to the single factor f a1a2a3 . In the Fortran program VBFNLO all permutations of boxes are combined to one box factor

1 µ1µ2µ3 Bµ1µ2µ3 (q , q , q , m )= B (q , q , q , m ) Φ,f 1 2 3 f 2 Φ,f 1 2 3 f µ2µ3µ1 h µ3µ1µ2 + BΦ,f (q2, q3, q1, mf )+ BΦ,f (q3, q1, q2, mf ) , (B.28) i which is totally antisymmetric in the gluon indices (qi,µi), i =1, 2, 3.

87 B.2.2 Fermion-box with -odd Higgs vertex CP The evaluation of the Dirac-trace for the permutation (1, 2, 3) of attached gluons with leads to VA

µ1µ2µ3 2 BA,f (q1, q2, q3, mf )=4 mf ( εµ3q1q2q3 gµ1µ2 εµ2q1q2q3 gµ1µ3 + εµ2µ3q2q3 qµ1 εµ2µ3q1q3 qµ1 − 1 − 2 h 1 2 2 + εµ2µ3q1q2 qµ + εµ1q1q2q3 gµ2µ3 + εµ1µ3q2q3 qµ εµ1µ3q1q2 qµ 3 1 − 3 εµ1µ2q2q3 qµ3 + εµ1µ2q1q3 qµ3 + εµ1µ2µ3q3 gµ1µ2 εµ1µ2µ3q2 gµ1µ3 − 1 2 − µ1µ2µ3q1 µ2µ3 µ1µ3q1q3 µ2 µ2 µ1µ2q1q2 µ3 + ε g + ε 2 q1 + q2 + ε 2(q1

µ3 µ3
 + q2 )+ q3 D0(q1, q2, q3, mf ) i εµ1µ2µ3q3 C (q + q , q , m ) εµ1µ2µ3q1 C (q , q + q , m ) − 0 1 2 3 f − 0 1 2 3 f µ2µ3q2q3 µ1 µ1µ3q1q3 µ2 +2 ε D (q1, q2, q3, mf )+2 ε D (q1, q2, q3, mf )

µ1µ2q1q2 µ3 +2 ε D (q1, q2, q3, mf ) . (B.29) )

The D0 and Dµ are four-point functions. Whereas the former denotes a scalar function, the latter can be expressed by the usual Passarino-Veltman decomposition [32] as

µ µ µ µ D (q1, q2, q3, mf )= q1 D11 + q2 D12 + q3 D13 . (B.30)

µ1 µ2 Note that after contraction with polarization vectors ǫ1 , ǫ2 and quark µ3 current J21 (3.9), the expression (B.29) still contains terms with factors (ǫ1 q1), (ǫ2 q2), (J21 q3) even though they vanish, since gluon polarization · µ · · µ vectors ǫi and momenta qi are perpendicular to each other and the quark current J21 is conserved. However, these terms are important for numerical gauge checks, where the corresponding gluon polarization vector is replaced 2 by its momentum. In this connection, the virtual gluon has a non-zero qi , and hence, these terms give finite contributions.

88 B.2.3 Fermion-box with -even Higgs vertex CP Following Ref. [2], the four-point function with -even Higgs vertex can be CP expressed in terms of few independent tensor structures

µ1µ2µ3 2 µ1µ2 µ3 BHSM ,h,H,f (q1, q2, q3, mf )=4 mf g q1 Ba(q1, q2, q3)

µ2µ3 µ1 n µ3µ1 µ2 + g q2 Ba(q2, q3, q1)+ g q3 Ba(q3, q1, q2) gµ2µ1 qµ3 B (q , q , q ) gµ1µ3 qµ2 B (q , q , q ) − 2 a 2 1 3 − 1 a 1 3 2 gµ3µ2 qµ1 B (q , q , q )+ qµ1 qµ2 qµ3 B (q , q , q ) − 3 a 3 2 1 3 3 1 b 1 2 3 µ1 µ2 µ3 µ1 µ2 µ3 + q2 q1 q1 Bb(q2, q3, q1)+ q2 q3 q2 Bb(q3, q1, q2) qµ1 qµ2 qµ3 B (q , q , q ) qµ1 qµ2 qµ3 B (q , q , q ) − 3 3 2 b 2 1 3 − 2 1 2 b 1 3 2 qµ1 qµ2 qµ3 B (q , q , q )+ qµ1 qµ2 qµ3 B (q , q , q ) − 3 1 1 b 3 2 1 2 3 1 c 1 2 3 qµ1 qµ2 qµ3 B (q , q , q ) , (B.31) − 3 1 2 c 2 1 3 with scalar functions o 1 B (q , q ,q )= q q D (q , q , q )+ D (q , q , q )+ D (q , q , q ) a 1 2 3 2 2 · 3 0 1 2 3 0 2 3 1 0 3 1 2 h i q q D (q , q , q )+ D (q , q , q ) D (q , q , q ) − 1 · 2 13 2 3 1 12 3 1 2 − 13 3 2 1 h i 4 D (q , q , q )+ D (q , q , q ) D (q , q , q ) − 313 2 3 1 312 3 1 2 − 313 3 2 1 h i C (q , q + q ) , (B.32) − 0 1 2 3 B (q , q ,q )= D (q , q , q )+ D (q , q , q ) D (q , q , q ) b 1 2 3 13 1 2 3 12 2 3 1 − 13 2 1 3

+4 D37(q1, q2, q3)+ D23(q1, q2, q3)+ D38(q2, q3, q1) h + D (q , q , q ) D (q , q , q ) D (q , q , q ) , (B.33) 26 2 3 1 − 39 2 1 3 − 23 2 1 3 1 i B (q , q ,q )= D (q , q , q )+ D (q , q , q )+ D (q , q , q ) c 1 2 3 −2 0 1 2 3 0 2 3 1 0 3 1 2 h i +4 D26(q1, q2, q3)+ D26(q2, q3, q1)+ D26(q3, q1, q2) h + D310(q1, q2, q3)+ D310(q2, q3, q1)+ D310(q3, q1, q2) . (B.34)

The coefficients Dij and Dijk originate from the Passarino-Veltmani tensor decomposition of Dµ, Dµ1µ2 and Dµ1µ2µ3 respectively. For more details see Ref. [2].

89 B.2.4 Sfermion-box with -even Higgs vertex CP

q q q q 2 k + q1 1 2 k + q23 1

µ1µ2µ3 µ1µ2µ3 k q k + q12 Bφ,1,f˜ k k + q3 Bφ,2,f˜ + 123

φ φ

q3 k + q123 q3 k

Figure B.7: Sfermionic box-loops with opposite fermion flow

The generic four-point functions for sfermionic box graphs with -even CP Higgs vertex and opposite loop momentum, shown in Fig. B.7, have the following expressions

4 µ1 µ2 µ1µ2µ3 d k 2 k + q1 2(k + q1)+ q2 BΦ,1,f˜ (q1, q2, q3, mf˜)= 2 iπ 2 2 2 2  k m (k + q1) m  Z − f˜ − f˜ h ih µ3 i 2(k + q12)+ q3 , (B.35) × 2 2 2 2 (k + q12) m (k + q123) m − f˜ − f˜ h 4 µih3 µi2 µ1µ2µ3 d k 2 k + q3 2(k + q3)+ q2 BΦ,2,f˜ (q1, q2, q3, mf˜)= 2 iπ 2 2 2 2  k m (k + q3) m  Z − f˜ − f˜ h ih µ1 i 2(k + q )+ q 23 1 . (B.36) × 2 2 2 2 (k + q23) m (k + q123) m − f˜ − f˜ h ih i After a shift of the loop momentum k k q q q , the relations → − − 1 − 2 − 3 of Eq. (B.25), (B.26) and (B.28) also holds for the sfermionic box factor. Evaluation of the numerator for the gluon permutation (1, 2, 3) yields

µ1µ2µ3 µ1µ2µ3 µ3 µ3 µ3 µ1µ2 BΦ,f˜ (q1, q2, q3, mf˜)=8 D +4 (2 q1 +2 q2 + q3 ) D

µ2 µ1µ3 µ2 µ1µ3 µ1 µ2µ3 µ2 µ2 µ3 +8 q1 D +4 q2 D +4 q1 D +2 (2 q1 + q2 ) 2 q1

µ3 µ3 µ1 µ1 µ3 µ3 µ3 µ2 µ1 µ2 µ3 +2 q2 + q3 D +2 q1 (2 q1 +2q2 + q3 ) D +4 q1 q1 D

µ1 µ2 µ3 µ1 µ2 µ2 µ3 µ3 µ3 +2 q1 q2 D
+ q1 (2 q1 + q2 )(2 q1 +2 q2 + q3 ) D0 . (B.37)

90 B.2.5 Sfermion-box with -even Higgs vertex and q˜qgg˜ - CP vertex

q q 1 k φ 1 k + q1234 φ

µ1µ2µ3µ4 k + q µ1µ2µ3µ4 q2 BP,φ,1,f˜ 1234 q2 BP,φ,2,f˜ k

k + q12 q4 k + q34 q4

q3 k + q123 q3 k + q4 Figure B.8: Sfermionic box-loops with opposite fermion flow and q˜qgg˜ vertex.

The cyclic permutation of the momentum set (q12, q3, q4) provides three differ- ent box contributions. The first contributions with opposite loop-momentum are depicted in Fig. B.8. All distinct box diagrams and their partners with opposite loop momentum flow are described now in more detail:

1) Permutation (q12, q3, q4)

4 µ1µ2 µ3 µ1µ2µ3µ4 d k g 2(k + q12)+ q3] BP,Φ,1,f˜ (q12, q3, q4, mf˜)= 2 iπ 2 2 2 2 k m (k + q12) m Z − f˜ − f˜ h ih µ4 i 2(k + q )+ q 123 4 , (B.38) × 2 2 2 2 (k + q123) m (k + q1234 ) m − f˜ − f˜ h 4 µih1µ2 µ4 i µ4µ3µ2µ1 d k g 2 k + q4] BP,Φ,2,f˜ (q4, q3, q12, mf˜)= 2 iπ 2 2 2 2 k m (k + q4) m Z − f˜ − f˜ h ih µ3 i 2(k + q4)+ q3 , (B.39) × 2 2 2 2 (k + q34) m (k + q1234) m − f˜ − f˜ h ih i 2) Permutation (q3, q4, q12)

4 µ1µ2 µ3 µ1µ2µ3µ4 d k g 2(k + q124)+ q3 BP,Φ,3,f˜ (q4, q12, q3, mf˜)= 2 iπ 2 2 2 2 k m (k + q4) m Z − f˜ − f˜ h ih i 91 µ4 2 k + q 4 , (B.40) × 2 2 2 2 (k + q124) m (k+ q1234) m − f˜ − f˜ h 4 µih1µ2 µ3 i µ4µ1µ2µ3 d k g 2 k + q3 BP,Φ,4,f˜ (q3, q12, q4, mf˜)= 2 iπ 2 2 2 2 k m (k + q3) m Z − f˜ − f˜ h ih µ4 i 2(k + q123)+ q4 , (B.41) × 2 2 2 2 (k + q123) m (k + q1234 ) m − f˜ − f˜ h ih i

3) Permutation (q4, q12, q3)

4 µ1µ2 µ3 µ3µ4µ1µ2 d k g 2 k + q3 BP,Φ,5,f˜ (q3, q4, q12, mf˜)= 2 iπ 2 2 2 2 k m (k + q3) m Z − f˜ − f˜ h ih µ4 i 2(k + q3)+ q4 , (B.42) × 2 2 2 2 (k + q4) m (k + q1234 ) m − f˜ − f˜ h 4 µ1µih2 µi4 µ1µ2µ3µ4 d k g 2(k + q12)+ q4 BP,Φ,6,f˜ (q12, q4, q3, mf˜)= 2 iπ 2 2 2 2 k m (k + q12) m Z − f˜ − f˜ h ih µ3 i 2(k + q124)+ q3 . (B.43) × 2 2 2 2 (k + q124) m (k + q1234 ) m − f˜ − f˜ h ih i Performing a shift of the loop momentum k k q in the box tensors → − − 1234 no. 2, 4, 6 gives

µ1µ2µ3µ4 µ4µ3µ2µ1 µ1µ2µ3µ4 B ˜ =+B ˜ B ˜ (q , q , q , m ˜) , (B.44) P,Φ,1,f P,Φ,2,f ≡ P,Φ,a,f 12 3 4 f µ1µ2µ3µ4 µ4µ3µ2µ1 µ1µ2µ3µ4 B ˜ =+B ˜ B ˜ (q , q , q , m ˜) , (B.45) P,Φ,3,f P,Φ,4,f ≡ P,Φ,b,f 3 4 12 f µ1µ2µ3µ4 µ4µ3µ2µ1 µ1µ2µ3µ4 B ˜ =+B ˜ B ˜ (q , q , q , m ˜) . (B.46) P,Φ,5,f P,Φ,6,f ≡ P,Φ,c,f 4 12 3 f

Furthermore, all diagrams have a pentagon-like color structure, which is de- noted by the index P . Due to the invariance property of the trace under cyclic permutations, the color structure is the same for all three contributions. For

92 a permutation such as (q12, q3, q4), it is given by

a1 a2 a3 a4 µ1µ2µ3µ4 a4 a3 a1 a2 µ4µ3µ2µ1 tr t , t t t BP,Φ,1,f˜ + tr t t t , t BP,Φ,2,f˜ h i h i = tr ta 1 ta2 ta3 ta4 + tr ta2 ta1 ta3 ta4 + tr ta4 t a3 ta1 ta2  a4 a3 a2 a1  µ1µ2µ3µ4    + tr t t t t BP,Φ,a (q12, q3, q4, mf˜) (B.47)  In the Fortran program VBFNLO all permutations are combined to one box factor

µ1µ2µ3µ4 µ1µ2µ3µ4 µ3µ4µ1µ2 µ4µ1µ2µ3 B (q , q , q , m ˜)= B + B + B (B.48) P,Φ,f˜ 12 3 4 f P,Φ,a,f˜ P,Φ,b,f˜ P,Φ,c,f˜ with

µ1µ2µ3µ4 BP,Φ,a,f˜ (q12, q3, q4, mf˜)

µ1µ2 µ3µ4 µ1µ2 µ3 µ3 µ3 µ4 µ1µ2 µ4 =4 g D +2 g 2 q1 +2 q2 + q3 D +2 g 2 q1

µ4 µ4 µ4 µ3 µ1µ2 µ3 µ3 µ3 µ4 +2 q2 +2 q3 + q4 D + g 2 q1 +2 q
2 + q3 2 q1

µ4 µ4 µ4 +2 q2 +2 q3 + q4
D0 ,
(B.49)
µ1µ2µ3µ4 BP,Φ,b,f˜ (q3, q4, q12, mf˜)

µ1µ2 µ3µ4 µ1µ2 µ3 µ3 µ3 µ3 µ4 =4 g D +2 g 2 q1 +2 q2 +2 q4 + q3 D

µ1µ2 µ4 µ3 µ1µ2 µ3 µ3 µ3 µ3 µ4 +2 g q4 D + g 2 q1 +2 q2 +2 q4 + q3
q4 D0 , (B.50)
µ1µ2µ3µ4 BP,Φ,c,f˜ (q4, q12, q3, mf˜)

µ1µ2 µ3µ4 µ1µ2 µ3 µ4 µ1µ2 µ4 µ4 µ3 =4 g D +2 g q3 D +2 g 2 q3 + q4 D

µ1µ2 µ3 µ4 µ4 + g q3 2 q3 + q4 D0 .
(B.51)

93 B.3 Five-point functions (Pentagons)

B.3.1 Fermion-pentagon with Higgs vertex

q1 q1 k q q2 k + q1 q2 + 234

k k + q1234 φ φ k q µ1µ2µ3µ4 k q µ1µ2µ3µ4 + 12 Pφ,1 + 34 Pφ,2

k + q1234 k q3 q3 k + q k + q123 4

q4 q4

Figure B.9: Five-point functions connected by charge-conjugation.

The two five-point functions connected by charge conjugation for the gluon permutation (1,2,3,4) have the following expressions d4k k + m P µ1µ2µ3µ4 (q , q , q , q , m )= tr f γµ1 Φ,1,f 1 2 3 4 f iπ2 k2 m2 Z " − f k +q + m k +q + m 1 f γµ2 12 f γµ3 × (k + q )2 m2 (k + q )2 m2 1 − f 12 − f k +q + m k +q + m 123 f γµ4 1234 f , (B.52) × (k + q )2 m2 (k + q )2 m2 VΦ 123 − f 1234 − f # d4k k + m P µ1µ2µ3µ4 (q , q , q , q , m )= tr f γµ4 Φ,2,f 1 2 3 4 f iπ2 k2 m2 Z " − f k +q + m k +q + m 4 f γµ3 34 f γµ2 × (k + q )2 m2 (k + q )2 m2 4 − f 34 − f k +q + m k +q + m 234 f γµ1 1234 f , (B.53) × (k + q )x2 m2 (k + q )2 m2 VΦ 234 − f 1234 − f # where q1, q2, q3 and q4 are outgoing momenta (qij = qi + qj and analog for qijk and qijkl). Using Furry’s theorem, one gets for the sum

a1 a2 a3 a4 a4 a3 a2 a1 µ1µ2µ3µ4 tr t t t t + tr t t t t PΦ,f (q1, q2, q3, q4, mf ) . (B.54)      94 In total, there exist 12 pentagons related by charge conjugation. Due to the length of the expressions for the fermionic pentagons, the tensorial structure will be described in a very brief way. For all pentagons, tensor reduction methods developed by Denner and Dittmaier [34, 35] were applied, which avoid the inversion of small Gram determinants, in particular for planar configurations of the Higgs and the two final state partons. Furthermore, all pentagons are UV- and IR-finite and are implemented as independent functions in the gluon fusion part GGFLO of the program VBFNLO [4]. The evaluation of the Dirac-trace with = γ contains Eµ1µ2 , Eµ, E0,Dµ, VA 5 D0 and C0 functions. The pentagon contribution for the -even Higgs bosons H , h and H with CP SM = 1 is composed of Eµ1µ2µ3µ4 , Eµ1µ2µ3 , Eµ1µ2 , Eµ1 , E0, Dµ1µ2 , Dµ1 VHSM,h,H and D0 functions.

B.3.2 Sfermion-pentagon with -even Higgs vertex CP

q1 q1 k q q2 k + q1 q2 + 234

k k + q1234 φ φ k q µ1µ2µ3µ4 k q µ1µ2µ3µ4 + 12 Pφ,1,f˜ + 34 Pφ,2,f˜

k + q1234 k q3 q3 k + q k + q123 4

q4 q4

Figure B.10: Sfermionic pentagon-loops with opposite fermion flow

The generic five-point functions for sfermionic pentagon graphs with - CP even Higgs vertex and opposite loop momentum, shown in Fig. B.10, have the following expressions

4 µ1µ2µ3µ4 d k µ1 µ2 P (q , q , q , q , m ˜)= 2 k + q 2(k + q )+ q Φ,1,f˜ 1 2 3 4 f iπ2 1 1 2 Z µ3  µ4   2(k + q12)+ q3 2(k + q123)+ q4 , (B.55) × 2 2 2 2 k  m ......  . . . (k + q1234)  m − f˜ − f˜ h ih ih ih ih i 95 4 µ1µ2µ3µ4 d k µ3 µ2 P (q , q , q , q , m ˜)= 2 k + q 2(k + q )+ q Φ,2,f˜ 1 2 3 4 f iπ2 4 4 3 Z µ1  µ1   2(k + q34)+ q2 2(k + q234)+ q1 . (B.56) × 2 2 2 2 k  m ......  . . . (k + q1234)  m − f˜ − f˜ h ih ih ih ih i The shorthand notation [. . .] denotes intermediate propagators. After a shift of the loop momentum, both Pentagons can be replaced by one expression with the color structure of Eq. (B.54). The evaluated expression of the sfermionic pentagon is given by

µ1µ2µ3µ4 P q , q , q , q , m ˜ h,H,f˜ 1 2 3 4 f µ1µ2µ3µ4 µ4
µ4 µ4 µ4 µ1µ2µ3 µ3 = 16E +8 2q1 +2q2 +2q3 + q4 E +8 2q1

µ3 µ3 µ1µ2µ4 µ2 µ2 µ1µ3µ4 µ1 µ2µ3µ4 µ3 +2q2 + q3 E +8 2q1 + q2 E
+8q1 E +4 2q1

µ3 µ3 µ4 µ4 µ4 µ4 µ1µ2 µ2 µ2 µ4 +2q2 + q3
2q1 +2q2 +2q3 + q
4 E +4 2q1 + q2 2q1

µ4 µ4 µ4 µ1µ3 µ1 µ4 µ4 µ4 µ4 µ2µ3 +2q2 +2q3
+ q4 E +4q1 2q1
+2q2 +2q3 + q4 E

µ2 µ2 µ3 µ3 µ3 µ1µ4 µ1 µ3 µ3 +4 2q1 + q2 2q1
+2q2 + q3 E +4q1 2q1 +2q2

µ3 µ2µ4 µ1 µ2 µ2 µ3µ4 µ2 µ2 µ3 µ3 + q3 E +4
q1 2q1 + q2 E
+2 2q1 + q2 2q1 +2q2

µ3 µ4 µ4 µ4 µ4 µ1 µ1 µ3 µ3 µ3 µ4 + q3
2q1 +2q2 +2q3 + q4
E +2q1 2q1 +2
q2 + q3 2q1

µ4 µ4 µ4 µ2 µ1 µ2 µ2 µ4 µ4 µ4 +2q2
+2q3 + q4 E +2q1
2q1 + q2 2q1 +2q2 +2q3

µ4 µ3 µ1 µ2 µ2 µ3 µ3 µ3 µ4 µ1 µ2 + q4 E +2q1 2
q1 + q2 2q1 +2q2 +
q3 E + q1 2q1

µ2 µ3 µ3 µ3 µ4 µ4 µ4 µ4 + q2
2q1 +2q2 + q3 2q
1 +2q2 +2q3 +
q4 E0 (B.57)


B.4 Six-point functions (Hexagons)

µ1µ2µ3µ4µ5 The hexagons HΦ,p (q1, q2, q3, q4, q5, mp) are computed using traditional methods, that is, computing the Feynman diagrams and giving the result in terms of tensor coefficient integrals Fij of six-point functions up to rank five for the -even case and up to rank three for the -odd case. In the CP CP process, some simplification has been used, the scalar products between of loop momenta and external momenta are rewritten in terms of propagators and some simplification can be done. Furthermore, tensor reduction methods of Denner and Dittmaier [34, 35] were used. In the hexagon subroutine also

96 appear Eij and some Dij functions. Moreover, all the gluons are kept off- shell to be able to attach to them some external current. For more details see Ref. [36]. The five external gluons give rise to 5! = 120 hexagons, which are of course proportional to 120 different color traces of the form

tr tai taj tak tal tam with i, j, k, l, m =1,..., 5   and i = j = k = l = m . (B.58) 6 6 6 6 In total there exist 60 hexagons related by charge conjugation. With the permutation (1,2,3,4,5) of the attached gluons the color structure for a sum of two hexagons reads as follows

tr ta1 ta2 ta3 ta4 ta5 tr ta5 ta4 ta3 ta2 ta1 −  1 2 34 5    Hµ µ µµ µ (q , q , q , q , q , m ) . (B.59) × Φ,p 1 2 3 4 5 p

97

Appendix C SU(N) identities

This appendix contains a summary of useful identities of the SU(N) algebra, which were used in processes, explained in former chapters. For more details see references [14, 54, 55, 56].

C.1 SU(N) tensors

δa1a2 =2tr ta1 ta2 , (C.1)

da1a2a3 =2tr ta1 ,
ta2 ta3 , (C.2)   f a1a2a3 =2itrta1 ta3 , ta2 . (C.3)    C.2 Traces of color generators

tr ta =0 , (C.4) 1 tr ta1ta2  = T δa1a2 = δa1a2 , (C.5) F 2   1 tr ta1 ta2 ta3 = da1a2a3 + if a1a2a3 , (C.6) 4   1
tr ta1 ta2 ta3 ta4 = δa1a2 δa3a4 4 N   1 + da1a2m + if a1a2m da3a4m + if a3a4m , (C.7) 8

99 1 tr ta1 ta2 ta3 ta4 ta5 = da1a2a3 + if a1a2a3 δa4a5 8 N   1
1 + δa1a2 da3a4a5 + if a3a4a5 + da1a2m + if a1a2m 8 N 16 dma3n + if ma3n dna4a5 + i
f na4a5 . (C.8)
×

C.3 Convolutions of da1a2a3 and f a1a2a3 with ta

a1a2m ma3n n d d ti2i1 = ta1 ta2 ta3 + ta2 ta1 ta3 + ta3 ta1 ta2 + ta3 ta2 ta1 i2i1 1 2 da1a2a3 δ δa1a2 ta3 ,
(C.9) − 3 i2i1 − 3 i2i1 a1a2m ma3n n f f ti2i1 = ta1 ta2 ta3 + ta2 ta1 ta3 + ta3 ta1 ta2 ta3 ta2 ta1 , (C.10) − − i2i1 a1a2m ma3n n
d f ti2i1 = i ta1 ta2 ta3 ta2 ta1 ta3 + ta3 ta1 ta2 + ta3 ta2 ta1 (C.11) − − i2i1 a1a2m ma3n n
f d ti2i1 = i ta1 ta2 ta3 + ta2 ta1 ta3 ta3 ta1 ta2 + ta3 ta2 ta1 − − i2i1 1 f a1a2a3 δ .
(C.12) − 3 i2i1

C.4 Jacobi identities

da1a2mf ma3a4 + da2a3mf ma1a4 + da3a1mf ma2a4 =0 , (C.13) f a1a2mf ma3a4 + f a1a4mf ma2a3 + f a1a3mf ma4a2 =0 . (C.14)

100 Appendix D QCD- and SQCD Vertices

This Appendix illustrates only the vertices of the QCD and SQCD used in former calculations. Further vertices, especially of the SQCD, can be looked up in Ref. [10].

D.1 QCD vertices

q ga µ a µ igSγ t (D.1) q

µ ga c g abc µν ρ νρ µ k gSf g (k p) + g (p q) ρ − − p h gb q + gρµ(q k)ν (D.2) − ν i

101 µ ν ig2 f abef cde gµρgνσ gµσgνρ − S − ga gb h + f acef bde gµνgρσ gµσgνρ
gc gd − + f adef bcegµρgνσ gµρgνσ
(D.3) ρ σ −
i D.2 SQCD vertices (only squarks)

q˜L,R/1,2 ga pi a µ µ igSt pi + pf (D.4) pf

q˜L,R/1,2

µ

a 2 a b µν q˜L,R/1,2 g igS t , t g (D.5) q˜ b L,R/1,2 g 

ν

102 Appendix E Effective Lagrangian

The calculation of Higgs + 2 jet and 3 jet via gluon fusion is quite involved, due to the fact, that already at leading order the Higgs boson is produced via a quark loop. For Higgs masses smaller than the threshold for the creation of a top-quark pair, mφ . 2 mt, and jet transverse energies smaller than the top-quark mass, p⊥ . mt [57] it is possible to replace different quark loop topologies by effective vertices [58]. In this connection low-energy theorems for Higgs boson interactions are used to relate amplitudes of two processes which differ by the insertions of a zero momentum Higgs boson [59, 60]. A direct comparison between loop and effective theory induced production of the -even and -odd Higgs boson with two jets is shown in plot E.1. CP CP The striking peak, which is absent in the effective limit, arises due to thresh- old enhancement at m 2 m . For that reason, finite width effects were H ≈ t included to control the behavior of this resonance. The threshold enhance- ment is smoother in the case of the -even Higgs boson, due to its large CP width, which is caused by additional contributions of decays to W and Z gauge bosons. For the -odd case those decays are forbidden at tree-level. CP The effective Lagrangian density for the -even Higgs boson H can be de- CP rived from the γγ H coupling, which is mediated by a triangle fermion − loop. Similar to that, one can derive the effective Lagrangian of the - CP odd Higgs boson A from the anomaly of the axial-vector current [61, 62]. Then, it is quite straightforward to generalize both Lagrangians to the non- Abelian SU(3) group. Hence, the effective theory is a powerful tool to exam- ine QCD-processes without enormous numerical effort. For more details see also Refs. [15, 63, 64]. The effective Lagrangians for both Higgs bosons read as

103 mA,H=120 GeV, IC 100 A, tan β=1 A, eff. th. H H, eff. th. [pb]

σ 10

1 100 200 300 400 500 600

mA [GeV]

Figure E.1: Comparison of cross sections of the -odd and -even CP CP Higgs coupling in both loop-induced and effective theory. Here, the inclusive cuts (IC) of Eq. (5.1) were applied.

= 0 + 0 Leff LH gg LA gg 1 α 1 α = s Ga Gµ1µ2aH + s Ga G˜µ1µ2aA (E.1) v 12π µ1µ2 v 8π µ1µ2 with 1 g2 1 1/2 G˜aµ1µ2 = ǫµ1µ2µ3µ4 Ga , α = s , = √2G , (E.2) 2 µ3µ4 s 4 π v F   a where Gαβ denotes the non-Abelian field strength tensor of the SU(3) gluon field, v the vacuum expectation value and GF the Fermi constant. The effective Lagrangians generate vertices involving two, three or four gluons and, of course, the Higgs bosons. Up to the box diagrams, the full loop and the effective theory can be related to each other very easily, because their color structure is equal.

104 Effective ggφ vertex triangle loops • ↔

The effective ggφ interactions for both Higgs bosons without couplings constants are [65] 1 T µ1µ2 (q , q )= δa1a2 gµ1µ2 q q qµ1 qµ2 and (E.3) H, eff 1 2 3 1 · 2 − 1 2 1
T µ1µ2 (q , q )= δa1a2 εµ1µ2µ3µ4 q q (E.4) A, eff 1 2 2 1, µ3 2, µ4 To involve full mass dependence, one has to replace the effective vertices a1a2 µ1µ2 by the expression δ Tφ (q1, q2, mt) (B.5) containing two charge- conjugated triangle loops

effective gggφ vertex box loops • ↔

The tensor structure of the effective gggφ interaction corresponds ex- actly to that of a three-gluon vertex (D.2)

Bµ1µ2µ3 (q , q , q )= f a1a2a3 gµ1µ2 (q q )µ3 + gµ2µ3 (q q )µ1 H, eff 1 2 3 2 − 1 3 − 2 h + gµ3µ1 (q q )µ2 . (E.5) 1 − 3 i a1a2a3 µ1µ2µ3 Inserting a box diagram denoted by f Bφ (q1, q2, q3, mt) (B.28) into the effective vertex brings the full mass dependence back. In total, however, there are six box graphs with different permutations of the external momenta. These can be reduced via Furry’s theorem [29] to three boxes with cyclicly permuted gluons.

105 effective ggggφ vertex pentagon loops • ↔

” ”

0 0 The effective ggggφ interaction with φ = HSM, h ,H is a copy of the four-gluon vertex (D.3)

P µ1µ2µ3µ4 (q , q ,q , q )= f a1a2mf a3a4m gµ1µ3 gµ2µ4 gµ1µ4 gµ2µ3 H, eff 1 2 3 4 − h + f a1a3mf a2a4m gµ1µ2 gµ3µ4 gµ1µ4 gµ2µ3
− + f a1a4mf a2a3mgµ1µ3 gµ2µ4 gµ1µ3 gµ2µ4
(E.6) −
i For the -odd Higgs boson A0, there exists no effective vertex. It van- CP ishes due to the anti-symmetry of the Levi-Civita symbol and structure constants and the Bose symmetry of the four attached gluons. A de- tailed proof is described in Ref. [3]. The non existence of this vertex can also be explained in a geometrical way. The four polarization vectors contain only space-like entries and hence span a three-dimensional sub- space of the Minkowski vector space. Due to the fact that the maximal number of basis vectors is three, the fourth vector has to be a linear combination of the basis vectors. For this case the contraction of the four polarization vectors with the Levi-Civita symbol is exactly zero. The full theory contains 4! = 24 pentagons. Furry’s theorem reduces the number of pentagons to 12. In chapter 3.3.3 it is shown that only six different color traces exist, which can be combined further to three

real-valued color factors ci (3.31). By means of c1 (3.32),

1 2 c = δa1a2 δa3a4 + da1a2mda3a4m f a1a2mf a3a4m , (E.7) 1 4 N −   one can see that the additional two terms spoil the simplicity of the color structure of a four-gluon vertex. Therefore, diagrams with a pen- tagon insertion have to be treated separately.

106 Appendix F gg → gggΦ

F.1 Remaining color factors for diagrams with pentagons

The momenta configuration of the gluon current (3.10) is denoted by squared brackets. Furthermore, the structure constants f are contracted via the index m with the ci of Eq. (3.31) using the following prescription

mapar aj ak al m mapar ci f = tr t t t t f with j,k,l,m,p,r =1,..., 5 and j = k = l = p = r . 6 6 6 6 Here the index i = 1, 2, 3 denotes the corresponding order of the color gen- erators inside the trace for a given ci. The remaining permutations are

permutation: 3, 4, 5, [1, 2] ◦  c f ma1a2 = i tr ta3 ta4 ta5 ta1 ta2 tr ta3 ta4 ta5 ta2 ta1 1 − −  + trta3 ta1 ta2 ta5 ta4  trta3 ta2 ta1 ta5 ta4  , (F.1) −  c f ma1a2 = i trta3 ta5 ta1 ta2 ta4  trta3 ta5 ta2 ta1 ta4  2 − −  + trta3 ta4 ta1 ta2 ta5  trta3 ta4 ta2 ta1 ta5  , (F.2) −  c f ma1a2 = i trta3 ta1 ta2 ta4 ta5  trta3 ta2 ta1 ta4 ta5  3 − −  + trta3 ta5 ta4 ta1 ta2  trta3 ta5 ta4 ta2 ta1  , (F.3) −     107 permutation: 2, 4, 5, [1, 3] ◦ c f ma1a3 = i tr ta2 ta4 ta 5 ta1 ta3 tr ta2 ta4 ta5 ta3 ta1 1 − −  + trta2 ta1 ta3 ta5 ta4  trta2 ta3 ta1 ta5 ta4  , (F.4) −  c f ma1a3 = i trta2 ta5 ta1 ta3 ta4  trta2 ta5 ta3 ta1 ta4  2 − −  + trta2 ta4 ta1 ta3 ta5  trta2 ta4 ta3 ta1 ta5  , (F.5) −  c f ma1a3 = i trta2 ta1 ta3 ta4 ta5  trta2 ta3 ta1 ta4 ta5  3 − −  + trta2 ta5 ta4 ta1 ta3  trta2 ta5 ta4 ta3 ta1  , (F.6) −  permutation: 2, 3, 5, [1, 4]    ◦ c f ma1a4 = i tr ta2 ta3 ta 5 ta1 ta4 tr ta2 ta3 ta5 ta4 ta1 1 − −  + trta2 ta1 ta4 ta5 ta3  trta2 ta4 ta1 ta5 ta3  , (F.7) −  c f ma1a4 = i trta2 ta5 ta1 ta4 ta3  trta2 ta5 ta4 ta1 ta3  2 − −  + trta2 ta3 ta1 ta4 ta5  trta2 ta3 ta4 ta1 ta5  , (F.8) −  c f ma1a4 = i trta2 ta1 ta4 ta3 ta5  trta2 ta4 ta1 ta3 ta5  3 − −  + trta2 ta5 ta3 ta1 ta4  trta2 ta5 ta3 ta4 ta1  , (F.9) −     permutation: (2, 3, 4, (1, 5) ◦ n o c f ma1a5 = i tr ta2 ta3 ta4 ta1 ta5 tr ta2 ta3 ta4 ta5 ta1 1 − −  + trta2 ta1 ta5 ta4 ta3  trta2 ta5 ta1 ta4 ta3  , (F.10) −  c f ma1a5 = i trta2 ta4 ta1 ta5 ta3  trta2 ta4 ta5 ta1 ta3  2 − −  + trta2 ta3 ta1 ta5 ta4  trta2 ta3 ta5 ta1 ta4  , (F.11) −  c f ma1a5 = i trta2 ta1 ta5 ta3 ta4  trta2 ta5 ta1 ta3 ta4  3 − −  + trta2 ta4 ta3 ta1 ta5  trta2 ta4 ta3 ta5 ta1  , (F.12) −     108 permutation: (1, 4, 5, (2, 3) ◦ n o c f ma2a3 = i tr ta1 ta4 ta5 ta2 ta3 tr ta1 ta4 ta5 ta3 ta2 1 − −  + trta1 ta2 ta3 ta5 ta4  trta1 ta3 ta2 ta5 ta4  , (F.13) −  c f ma2a3 = i trta1 ta5 ta2 ta3 ta4  trta1 ta5 ta3 ta2 ta4  2 − −  + trta1 ta4 ta2 ta3 ta5  trta1 ta4 ta3 ta2 ta5  , (F.14) −  c f ma2a3 = i trta1 ta2 ta3 ta4 ta5  trta1 ta3 ta2 ta4 ta5  3 − −  + trta1 ta5 ta4 ta2 ta3  trta1 ta5 ta4 ta3 ta2  , (F.15) −  permutation: (1, 3, 5, (2, 4)    ◦ n o c f ma2a4 = i tr ta1 ta3 ta5 ta2 ta4 tr ta1 ta3 ta5 ta4 ta2 1 − −  + trta1 ta2 ta4 ta5 ta3  trta1 ta4 ta2 ta5 ta3  , (F.16) −  c f ma2a4 = i trta1 ta5 ta2 ta4 ta3  trta1 ta5 ta4 ta2 ta3  2 − −  + trta1 ta3 ta2 ta4 ta5  trta1 ta3 ta4 ta2 ta5  , (F.17) −  c f ma2a4 = i trta1 ta2 ta4 ta3 ta5  trta1 ta4 ta2 ta3 ta5  3 − −  + trta1 ta5 ta3 ta2 ta4  trta1 ta5 ta3 ta4 ta2  , (F.18) −  permutation: (1, 3, 4, (2, 5)    ◦ n o c f ma2a5 = i tr ta1 ta3 ta4 ta2 ta5 tr ta1 ta3 ta4 ta5 ta2 1 − −  + trta1 ta2 ta5 ta4 ta3  trta1 ta5 ta2 ta4 ta3  , (F.19) −  c f ma2a5 = i trta1 ta4 ta2 ta5 ta3  trta1 ta4 ta5 ta2 ta3  2 − −  + trta1 ta3 ta2 ta5 ta4  trta1 ta3 ta5 ta2 ta4  , (F.20) −  c f ma2a5 = i trta1 ta2 ta5 ta3 ta4  trta1 ta5 ta2 ta3 ta4  3 − −  + trta1 ta4 ta3 ta2 ta5  trta1 ta4 ta3 ta5 ta2  , (F.21) −     109 permutation: (1, 2, 5, (3, 4) ◦ n o c f ma3a4 = i tr ta1 ta2 ta5 ta3 ta4 tr ta1 ta2 ta5 ta4 ta3 1 − −  + trta1 ta3 ta4 ta5 ta2  trta1 ta4 ta3 ta5 ta2  , (F.22) −  c f ma3a4 = i trta1 ta5 ta3 ta4 ta2  trta1 ta5 ta4 ta3 ta2  2 − −  + trta1 ta2 ta3 ta4 ta5  trta1 ta2 ta4 ta3 ta5  , (F.23) −  c f ma3a4 = i trta1 ta3 ta4 ta2 ta5  trta1 ta4 ta3 ta2 ta5  3 − −  + trta1 ta5 ta2 ta3 ta4  trta1 ta5 ta2 ta4 ta3  , (F.24) −     permutation: (1, 2, 4, (3, 5) ◦ n o c f ma3a5 = i tr ta1 ta2 ta4 ta3 ta5 tr ta1 ta2 ta4 ta5 ta3 1 − −  + trta1 ta3 ta5 ta4 ta2  trta1 ta5 ta3 ta4 ta2  , (F.25) −  c f ma3a5 = i trta1 ta4 ta3 ta5 ta2  trta1 ta4 ta5 ta3 ta2  2 − −  + trta1 ta2 ta3 ta5 ta4  trta1 ta2 ta5 ta3 ta4  , (F.26) −  c f ma3a5 = i trta1 ta3 ta5 ta2 ta4  trta1 ta5 ta3 ta2 ta4  3 − −  + trta1 ta4 ta2 ta3 ta5  trta1 ta4 ta2 ta5 ta3  . (F.27) −    

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116 Acknowledgements

My special thanks go to Prof. Dr. Dieter Zeppenfeld, for giving me the op- portunity to work on this very interesting topic. I am particularly grateful for his advice, his interest in my work, and also for support in many respects. Special thanks go to Prof. Dr. Margarete M¨uhlleitner for the short-term acceptance of proofreading. Also special thanks go to Gunnar Kl¨amke for the great and important co- operation during my PhD thesis, to my room mates Christian Schappacher and Francisco Campanario for many important discussions (in English), the willingness to help and for the fantastic atmosphere, to which also our former room mate Luca D’Errico contributed a lot. Therefore also special thanks to Luca. Special thanks to Christian Schappacher and Francisco Campanario for care- fully reading of my Diss. and for giving me helpful advices. Special thanks to Simon Pl¨atzer for the help in technical questions about computer and other stuff and for interesting discussions. Special thanks to Malin Sj¨odahl for the powerful color algebra mathematica package. Special thanks to Anastasija Bierweiler, Philipp Maierh¨ofer, Cornelius Rampf and Gunnar Kl¨amke for interesting discussions during the lunch time and leisure time. I want to thank the Graduiertenkolleg ”Hochenergiephysik und Teilchenas- trophysik” and the Landesgraduiertenkolleg for its financial support. I want to thank all members of the ITP for the fantastic atmosphere and helpfulness. I am deeply grateful to my parents and grandparents for the tremendous encouragement during the whole study.

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